The Proof is in the Pudding The Changing Nature of Mathematical Proo
Krantz’s book covers the full history and evolution of the proof concept. The notion of rigorous thinking has evolved over time, and this book documents that development. It gives examples both of decisive developments in the techn
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Steven G. Krantz
The Proof is in the Pudding The Changing Nature of Mathematical Proof
Steven G. Krantz Department of Mathematics Washington University in St. Louis St. Louis, MO 63130 USA [email protected]
ISBN 978-0-387-48908-7 e-ISBN 978-0-387-48744-1 DOI 10.1007/978-0-387-48744-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011928557 Mathematics Subject Classification (2010): 01-XX, 03-XX, 65-XX, 97A30 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Jerry Lyons, mentor and friend
0 Preface The title of this book is not entirely frivolous. There are many who will claim that the correct aphorism is “The proof of the pudding is in the eating.”—that it makes no sense to say, “The proof is in the pudding.” Yet people say it all the time, and the intended meaning is usually clear. So it is with mathematical proof. A proof in mathematics is a psychological device for convincing some person, or some audience, that a certain mathematical assertion is true. The structure, and the language used, in formulating such a proof will be a product of the person creating it; but it also must be tailored to the audience that will be receiving it and evaluating it. Thus there is no “unique” or “right” or “best” proof of any given result. A proof is part of a situational ethic: situations change, mathematical values and standards develop and evolve, and thus the very way that mathematics is done will alter and grow. This is a book about the changing and growing nature of mathematical proof. In the earliest days of mathematics, “truths” were established heuristically and/or empirically. There was a heavy emphasis on calculation. There was almost no theory, no formalism, and there was little in the way of mathematical notation as we know it today. Those who wanted to consider mathematical questions were thereby hindered: they had difficulty expressing their thoughts. They had particular trouble formulating general statements about mathematical ideas. Thus it was virtually impossible for them to state theorems and prove them. Although there are some indications of proofs even on ancient Babylonian tablets (such as Plimpton 322) from 1800 BCE, it seems that it is in ancient Greece
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