Mathematical Concepts
The main intention of this book is to describe and develop the conceptual, structural and abstract thinking of mathematics. Specific mathematical structures are used to illustrate the conceptual approach; providing a deeper insight into mutual relationshi
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Mathematical Concepts
Mathematical Concepts
Jürgen Jost
Mathematical Concepts
123
Jürgen Jost Max Planck Institute for Mathematics in the Sciences Max Planck Society Leipzig Germany
ISBN 978-3-319-20435-2 DOI 10.1007/978-3-319-20436-9
ISBN 978-3-319-20436-9
(eBook)
Library of Congress Control Number: 2015943831 AMS Codes: 00A05, 00A06, 01A65, 03-02, 03B45, 03G30, 18B25, 18-XX, 18-02, 18Axx, 18Fxx, 18Gxx, 20A05, 20B05, 20C05, 06Bxx, 08Axx, 14-XX, 14A15, 51K10, 53B05, 53B20, 53C21, 54A05, 55Nxx, 55P10, 55U10, 58A05, 92B99 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)
Mathematics translates concepts into formalisms and applies those formalisms to derive insights that are usually not amenable to a less formal analysis.
Preface
This book provides an introduction to and an overview of the most important concepts and structures of modern mathematics. Therefore, I try to motivate and explain those concepts and structures and illustrate them with examples, but I often do not provide the complete proofs of the basic results. Therefore, this book has a broader scope and is less complete and detailed than standard mathematical textbooks. Its main intention is to describe and develop the conceptual, structural, and abstract thinking of mathematics. Specific mathematical structures then illustrate that conceptual approach, and their mutual relationships and their abstract common features should also provide deeper insight into each of them. The book thus could be used 1. simply as an overview of the panorama of mathematical structures and the relations between them, to be supplemented by more detailed texts whenever you want to acquire a working knowledge of some structure, 2. by itself as a first introduction to abstract mathematics, 3.
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