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Springer Science+Business Media, LLC

Yu. I. Manin

Mathematics andPhysics Translated by Ann and Neal Koblitz

1981

Springer Science+Business Media, LLC

Author Yu. 1. Hanin Steklov Hathematical Institute Vavi lova 42 117966, GSP-1 Hoscow USSR

Library of Congress Cataloglng in Publication Data Hanin, IU. 1. Hathematics and Physics. (Progress in Physics ; 3) fizika. Translation of: Hatematika 2. Physics. 1. Title. 1. Hathematics--196111. Series: Progress In Physics (Boston, Mass.); 3.

QA37.2.M3413

530.1'5

81-1319 AACR2

CIP- Kurztitelaufnahme der Deutschen Bibliothek Hanin, Jurij 1.: Hathematics and Physics 1 Yu. 1. Manin. Translated by Ann and Neal Koblitz Boston: Basel: Stuttgart: Birkh~user, 1981 (Progress In Physics ; 3) ISBN 978-1~84-8 NE:GT Orlginally published In Novoye v Zhizny, Naukye, Tekhnlke, Seriva Hatematika, Kibernetika, Hoscow: Znaniye, No.12, 1979. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopylng, recordlng or otherwise, without prior permisslon of the copyright owner. © Springer Science+Business Media New York 1981

Originally published by Bi rkhl:luse r Boston in 1981 Softcover re print of the hardcover 1st edition 1981 ISBN 978-1-4899-6784-8 DOI 10.1007/978-1-4899-6782-4

ISBN 978-1-4899-6782-4 (eBook)

CONTENTS

FOREWORD

1.

A BIRD'S EYE VIEW OF MATHEMATICS

ix 1

Mathernatical Truth Sets Higher-Dimensional Space and the Idea of Linearity Measurernents in a Linear Space Nonlinearity and Curvature Sorne Novelties Sets, Forrnulas, and the Divided Brain 2.

PHYSICAL QUANTITIES, DI~ffiNSIONS AND CONSTANTS: The Source of Nurobers in Physics

35

The Spectrurn of a Scalar Quantity Physical Law, Dimension and Sirnilarity The Group of Dimensions of Newtonian Mechanics Scale Invariance Planck Units and the Problern of a Unified Physical Theory Classification of Physical Constants 3.

A DROP OF MILK: Observer, Observation, Observable and Unobservable

52

Isolated systerns Principles of Quantum Description Phase Space Phase Curves Observation: Ovens, Filters and Quantum JumpsCombination of Quantum Systems

4.

SPACE-TIME AS A PHYSICAL SYSTEM

71

The N.inkowski World Curved World Spinors, Twistors and the Cornplex World Space-Tirne, Gravitation and Quantum r1echanics

5.

ACTION AND SYMf.ffiTRY

88

The connection between mathematics and physics .•.

Is it only that physicists

talk in the language of mathematics? It is more.

This book describes how

mathematics associates to some important physical abstractions (modelsl its own mental constructions, which are far removed from the direct impressions of experience and physical experiment.

vii

FOREWORD

There is a story about how a certain well-known rnathernatician would begin his sophornore course in logic. the science of laws of thought," he would declairn.

"Logic is "Now I

rnust tell you what science is, what law is, and what thought is.

But I will not explain what 'of' rneans." On undertaking the task of wr