Explanation and Proof in Mathematics Philosophical and Educational P
In the four decades since Imre Lakatos declared mathematics a "quasi-empirical science," increasing attention has been paid to the process of proof and argumentation in the field -- a development paralleled by the rise of computer technology and the mount
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Explanation and Proof in Mathematics Philosophical and Educational Perspectives
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Explanation and Proof in Mathematics
Gila Hanna Hans Niels Jahnke Helmut Pulte ●
Editors
Explanation and Proof in Mathematics Philosophical and Educational Perspectives
Editors Gila Hanna Ontario Institute for Studies in Education (OISE) University of Toronto Toronto ON, Canada [email protected]
Hans Niels Jahnke Department of Mathematics University of Duisburg-Essen Essen Germany [email protected]
Helmut Pulte Ruhr-Universität Bochum Germany [email protected]
ISBN 978-1-4419-0575-8 e-ISBN 978-1-4419-0576-5 DOI 10.1007/978-1-4419-0576-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009933111 © Springer Science+Business Media, LLC 2010 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)
Contents
1 Introduction................................................................................................
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Part I Reflections on the Nature and Teaching of Proof 2 The Conjoint Origin of Proof and Theoretical Physics.......................... Hans Niels Jahnke
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3 Lakatos, Lakoff and Núñez: Towards a Satisfactory Definition of Continuity............................................................................. Teun Koetsier
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4 Preaxiomatic Mathematical Reasoning: An Algebraic Approach............................................................................. Mary Leng
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5 Completions, Constructions, and Corollaries......................................... Thomas Mormann 6 Authoritarian Versus Authoritative Teaching: Polya and Lakatos...................................................................................... Brendan Larvor 7 Proofs as Bearers of Mathematical Knowledge...................................... Gila Hanna and Ed Barbeau
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71 85
8 Mathematicians’ Individual Criteria for Accepting Theorems and Proofs: An Empirical Approach..................................... 101 Aiso Heinze
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Contents
Part II Proof and Cognitive Development 9 Bridging Knowing and Proving in Mathematics: A Didactical Perspective.......................................................................... 115 Nicolas Balacheff 10 The Long-Term Cognitive Development of Reasoning and Proof..........
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