Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Leo Corry
Modern Algebra and the Rise of Mathematical Structures
Second revised edition
Springer Basel AG
Author's address:
Dr. Leo Corry
The Cohn Institute for the History
and Philosophy of Science and Ideas Tel Aviv University
Ramat Aviv 69978 Israel
The first edition of this book was published in 1996 in the series Science Networks - Historical
Studies VoI. 17.
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C. U.SA
Bibliographic information published by Die Deutsche Bibliothek:
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; deatailed bibliographic data is available in the Internet at <http://dnb.ddb.de>
ISBN 978-3-7643-7002-2
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustra-tions, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained.
© 2004 Springer Basel AG Originally published by Birkhăuser Verlag, Basel - Boston - Berlin in 2004
Cover design: Micha Lotrovsky, CH-4106 Therwil, Switzerland
Printed on acid-free paper produced from chlorine-free pulp. TCF 00
ISBN 978-3-7643-7002-2 ISBN 978-3-0348-7917-0 (eBook) DOI 10.1007/978-3-0348-7917-0
987654321 www.birkhasuer-science.com
To the memory ofmy father,Ricardo Corry 2., ?"t
Preface to the second edition
Not many recent books on the history of mathematics have seen a second edition, I believe. Thus, I was greatly surprised and pleased when the editors ofScience Networks approached me, more than a year ago, asking if I would beinterested in publishing a second edition of mine. I was eager to accept, ofcourse, though in the end it took me much longer to complete the task than Ihad initially assumed.Since its publication, I have received many positive reactions about the
book, and I hope that it has had some significant impact on the current historiography ofmathematics. In the present edition, I have not made considerablechanges in the overall conception or in the details of the text, except wherenecessary. I have tried indeed to update the footnotes, by incorporating, wherever possible, references to relevant works that have appeared over the lastfive years. I have also made a great effort, hopefully to a certain degree of success, to simplify and improve the prose of some linguistically intricate passages, particularly in Chapter 2.Friends and colleagues have called my attention to many typos and other
minor errors in the first edition. I am grateful to them all. Some went even further and were kind enough to provide me with a detailed list of those placesthat needed to be corrected. I thank especially Jose Ferreiros, Colin McLarty,and John Allen, for their attentive reading and useful comments. I owe anenormous debt to Walter Purkert for systematically indicating problems withsome German quotations scattered throughout the text. Norbert Schappacherprovided me with new, important information on the work ofRudolf Fueter onnumber theory. Miriam Greenfield dispelled some of my doubts concerningthe English prose of the text. If there still remain some errors, especially typosor any kind of language oddities, in the present edition, I take full responsibility for them.
Preface to the first edition
This book grew out of a doctoral dissertation submitted to Tel Aviv Universityin 1990. Initially, I had intended to concentrate on a detailed account of the origins and early development of category theory. Although category theory isnot usually included under the label of "metamathematics", it is obviously ametamathematical theory in the etymological sense of the word. Much asproof theory, for instance, is a mathematical theory whose subject matter is awell-determined aspect ofmathematical knowledge and of mathematical practice-namely, mathematical proof-so does category theory involve anattempt to characterize and analyze, from the perspective of an elaboratemathematical theory, the organization of mathematical knowledge within different disciplines as well as the interconnections among those disciplines.Many disciplines usually included under the heading of "metamathematics"have been the object of considerable historical and philosophical research; category theory has until now received too little attention.Metamathematical theories are particularly interesting subjects for histor
ical and philosophical research, since they evince a peculiar trait of mathematics' namely, the possibility of discussing certain meta-disciplinary issueswithin the body of the discipline itself. Proof theory, which is a rather moderndiscipline, set out to elucidate in strict mathematical terms what has been aconstitutive element of mathematical knowledge at least since the time of theGreeks. Category theory, on the other hand, considers a much more recentidea-the idea of a mathematical structure. During a considerable portion ofthe present century mathematics has been perceived as a science of structures.Students have made their way in the discipline essentially by becoming gradually acquainted with the different structures that constitute contemporarymathematics. Along the way, they learn similar conceptual schemes and conceptual tools which are equally applied in the study of diverse structures. Category theory may well be characterized as the mathematical theory that seeksa systematic analysis of these different structures, and the recurring mathematical phenomena that arise in them.
Category theory was not the first attempt to formalize the idea of a mathematical structure. An well-know, earlier, one was advanced by Nicolas Bourbaki, a group of mathematicians whose name is typically associated with theidea of mathematical structure. Bourbaki formulated a theory of structures aspart of a multi-volume treatise, and one often reads that category theory constitutes an improvement of Bourbaki's theory. Less well-known is a thirdattempt, advanced by Oystein Ore from around 1935, to provide a general concept of algebraic structure in terms of lattice-theoretical ideas. Ore's programstressed the fact that a general theory of structures must ignore the existenceof elements in each individual mathematical system, and that instead it mustconcentrate on the inter-relations among them. This line of thought, more systematically developed, emerged as a leading principle of category as well.The existence of these three theories, together with the well-known per
vasiveness of the idea of structure in contemporary mathematics, suggestsor at least so it seems to me-that in order to describe the origins of categorytheory, one could start from the stage at which the conception of mathematicsunderstood in terms of mathematical structure was widely adopted in mathematics. From this vantage point of view, one should then discuss the rise anddevelopment of Ore's and Bourbaki's theories, and their relationship, if any,to the rise in category theory.I had originally planned to follow this line of argumentation. However, it
soon appeared to me that the very idea of mathematical structure was a ratherfuzzy one, and that the parallel between category theory and other metamathematical theories cannot shed much light on the deeper questions involved. Itis not only that mathematical structures are a much more recent feature ofmathematical knowledge and practice than, say, mathematical proofs. Theidea of a mathematical structure, in spite of it ubiquity in twentieth centurymathematics, is an idea whose nature, meaning and role are not clearly understood and have seldom been systematically discussed. Thus, the more onestudies texts in which the term "mathematical structure" appears, the clearer itbecomes that this notion has been used in different contexts by different persons, with different meanings in mind, or with no clear meaning at all.This being the case, I decided to broaden the initial scope of my research
in order to try to answer some more general historical questions, such as: Howcan the structural approach to mathematics be characterized? What is a mathematical structure? When, and as a result of which processes, did structures
begin to be adopted in mathematical research and practice? When did mathematicians become conscious of this adoption? How did they respond to it?The present book is an attempt to answer some of these questions, at least
partially, and also to suggest a framework for a more comprehensive study ofthem. Thus, the book is divided into two parts. Part One describes the development of ideal theory from Richard Dedekind to Emmy Noether. This particular development transformed a theory, initially conceived in the context ofaccounting for factorization properties of algebraic numbers, into a paradigmatic structural theory of modern algebra. I have tried to ascribe a more precise meaning to the term "structural approach" in this development, whileexplaining the specific differences between Dedekind's and Noether's theoriesthat spell out why the former is "less structural" and the latter a "more structural" mathematical theory. In particular, I have tried to show that the structural approach to mathematics did not come about simply because it becameclear that mathematical concepts can be formulated abstractly. This was certainly an important factor in the overall process. However, it was also necessary for mathematicians to reflect on the interesting questions to be askedabout such questions as well as on the legitimate, interesting answers to beexpected when addressing those questions while using particular kinds of conceptual tools. I have chosen to pursue here these questions in the context ofideal theory. The rise of structural approach to mathematics in generalinvolved a much more complex process, that took place in different disciplinesat different paces, motivated by diverse kinds of mathematical concerns.Part Two deals with three different mathematical theories that may be
understood as attempts to elucidate the concept of mathematical structure,namely, those advanced by Ore and Bourbaki, together with category theory.Drawing on the developments presented in Part One. I describe here the rootsof three attempts, and their historical and conceptual interrelations.My initial plan for dealing with the rise of category theory was thus essen
tially transformed: I describe in much less detail than originally thought thedirect roots and the initial stages of category theory itself. On the other hand,I have attempted to provide a wider perspective from which the detailedaccount should be considered. Such a detailed account thus remains for afuture undertaking.
In writing this work-first as a doctoral thesis and later as a book-I havereceived help and support from many persons to whom I am deeply grateful.First and foremost, lowe a great debt to Sabetai Unguru, who introduced meinto the study of the history of mathematics and supervised the writing of mydissertation. His wise guidance and constant encouragement have been a stimulating source of support.
In the long and winding process of transforming my dissertation into a finished book, I have enjoyed the invaluable editorial help of David Rowe. Hegenerously put at my disposal much more of his time, professional expertise,and of his uncompromising critical abilities than I could have deservedlyasked for. His many suggestions and criticisms made me rethink considerableportions of the book, as well as its overall structure. The present form of thebook, however, should not be taken in any way as representing his own viewson the issues discussed. In fact, he explicitly expressed his disagreement withthe options put forward in some sections of the book. Still, where he agreedand where he disagreed, his comments always helped me to improve my arguments and presentations.lowe special thanks to Professor Saunders Mac Lane for insightful
remarks on a draft of chapters 8 and 9, as well as for useful material he sent tome, and to Professor Samuel Eilenberg, for an illuminating conversation in TelAviv in 1986, on the occasion of his being awarded the Wolf Prize.Several people have been kind enough to read and comment upon por
tions of earlier versions of the manuscript. I thank them all for their helpfulremarks: Liliane Beaulieu, Pierre Cartier, Newton C.A da Costa, Andree Ch.Ehresmann, Jose Ferreir6s, Catherine Goldstein, Ivor Grattan-Guinness, RalfHaubrich, Giorgio Israel, Colin Mclarty, Herbert Mehrtens.I have benefited from information and opinions kindly communicated to
me, in answer to my letters, by Professors Garret Birkhoff, the late Jean Dieudonnee, Harold Dorwart, Harold Edwards, Solomon Feferman, Peter Hilton,Jean-Pierre Serre and Pierre Samuel.Naturally, none of the persons mentioned above should be held responsi
ble for the mistakes, excesses or shortcomings that the reader may still find inthe book.Professor Andree Ch.Ehresmann, at Amiens, kindly placed at my disposal
original, unpublished material belonging to the estate of her late husband, Professor Charles Ehresmann. I thank her very much for her help, and for havinggranted me permission to quote from these documents.
Portions of Chapter 7 and 8 appeared originally in my article "NicolasBourbaki and the Concept ofMathematical Structure", Synthese 92, 1992, pp.315-348. I thank Kluwer Academic Publishers for allowing their reproduction(with considerable changes) here.I also thank Dr.Helmuth Rohlfing from the Handschriftenabteilung, Nied
ersachsiche Staats- und Universitatbibliothek Gottingen, for his helpful adviseand for permission to quote from David Hilbert's Nachlass (Cod.Ms.Hilbert558, Mechanik, WS 1898-99). Likewise, I thank Mr. Mathees, librarian of theMathematisches Institut, Universitat Gottingen, for permission to quote froma manuscript of Hilbert's lecturers Logische Principien des MathematischesDenkens, SS 1905.Ayelet Raemer helped me revising the English prose of the final version.
I enormously appreciate her kind help. Any Hispanisms, Hebraisms and otherlinguistic oddities that may have remained in the text are all of my exclusivechoice and responsibility. All translations into English that appear in the textare mine, unless otherwise indicated.In preparing the camera-ready copy of the book, I benefited from the kind
help of Yonatan Kaplun, Ido Yavetz and Studio Itzuvnik, and from the technical guidance of Doris Womer from Birkhauser Verlag. I thank them all.Most of the research and the writing of this book was done in the environs
of the Cohn Institute for the History and Philosophy if Science and Ideas, TelAviv University. During the years that I have been connected with the CohnInstitute I have had the opportunity to meet many people, who contributed toshaping my own ideas and approach to the history of mathematics in ways thatcannot be acknowledge in footnotes. I wish to thank especially Yehuda Elkanafor constant support and encouragement, and Gabriela Williams for herdevoted help. Shmuel Rosset, of the School ofMathematics, Tel Aviv University, has been faithful interlocutor on mathematical, as well as on other issuesover many years. His critical attitude towards my work has been of great help.Last, but certainly not least, I want to thank all those who constitute my
non-academic environment, and who have provided over the years a congenialatmosphere for carrying out my work: all my friends in Kibbutz Nirim, myfamily, and, above all, my dear wife Efrat.
Tel Aviv - Nirim, October 1995
Table of Contents
Introduction: Structures in Mathematics 1
Part One: Structures in the Images of Mathematics 13
Chapter 1 Structures in Algebra: Changing Images 19
1.1 Jordan and Holder: Two Versions of a Theorem 221.2 Heinrich Weber: Lehrbuch der Algebra 331.3 Bartel L. van der Waerden: Moderne Algebra .431.4 Other Textbooks of Algebra in the 1920s 54
Chapter 2 Richard Dedekind: Numbers and Ideals 64
2.1 Lectures on Galois Theory 762.1 Algebraic Number Theory 81
2.2.1 Ideal Prime Numbers 812.2.2 Theory ofIdeals: The First Version (1871) 932.2.3 Later Versions 1042.2.4 TheLastVersion ll02.2.5 Additional Contexts 118
2.3 Ideals and Dualgruppen 1212.4 Dedekind and the Structural Image of Algebra 129
Chapter 3 David Hilbert: Algebra and Axiomatics 137
3.1 Algebraic Invariants 1383.2 Algebraic Number Theory 1473.2 Hilbert's Axiomatic Approach 1543.4 Hilbert and the Structural Image of Algebra 1683.5 Postulational Analysis in the USA 172
Contents
Chapter 4 Concrete and Abstract:Numbers, Polynomials, Rings 183
4.1 Kurt Hensel: Theory of p-adic Numbers 1844.2 Ernst Steinitz: Algebraische Theorie der Korper 1924.3 Alfred Loewy: Lehrbuch der Algebra 1964.4 Abraham Fraenkel: Axioms for p-adic Systems 2014.5 Abraham Fraenkel: Abstract Theory of Rings 2074.6 Ideals and Abstract Rings after Fraenkel 2134.7 Polynomials and their Decompositions 214
Chapter 5 Emmy Noether: Ideals and Structures 220
5.1 Early Works 2215,2 Idealtheorie in Ringbereichen 2255.3 Abstrakter Aufbau der Idealtheorie 2375.4 Later Works 2445.5 Emmy Noether and the Structural Image of Algebra 247
Part Two: Structures in the Body of Mathematics 253
Chapter 6 Oystein are: Algebraic Structures 259
6.1 Decomposition Theorems and Algebraic Structures 2616.2 Non-Commutative Polynomials and Algebraic Structure 2636.3 Structures and Lattices 2676.4 Structures in Action 2756.5 Universal Algebra, Model Theory, Boolean Algebras 2796.6 Ore's Structures and the Structural Image of Algebra 286
Contents
Chapter 7 Nicolas Bourbaki: Theory of Structures 289
7.1 The Myth 2917.2 Structures and Mathematics 3017.3 Structures and the Body ofMathematics 311
7.3.1 Set Theory 3127.3.2 Algebra 3227.3.3 General Topology 3257.3.4 Commutative Algebra 327
7.4 Structures and the Structural Image of Mathematics 329
Chapter 8 Category Theory: Early Stages 339
8.1 Category Theory: Basic Concepts 3408.2 Category Theory: A Theory of Structures 3448.3 Category Theory: Early Works 3518.4 Category Theory: Some Contributions 3648.5 Category Theory and Bourbaki 372
Chapter 9 Categories and Images of Mathematics 380
9.1 Categories and the Structural Image of Mathematics 3809.2 Categories and the Essence of Mathematics 3869.3 What is Algebra and what has it been in History? 395
Bibliography
Author Index
Subject Index
399
433
441
Related Documents
