Figures of Thought: Mathematics and Mathematical Texts

Figures of Thought

Figures of Thought Mathematics and Mathematical Texts

David Reed

London and New York

First published 1995 by Routledge 11 New Fetter Lane, London EC4P 4EE This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” Simultaneously published in the USA and Canada by Routledge 29 West 35th Street, New York, NY 10001 © 1995 David Reed All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress ISBN 0-203-16828-3 Master e-book ISBN

ISBN 0-203-26347-2 (Adobe eReader Format) ISBN 0-415-08146-7 (Print Edition)

To Zena
without whom not


Contents

List of figures Preface Introduction

Part I The subject matter of geometry in Euclid, Descartes and Hilbert 1

viii ix x

1

The opening of the Elements

3

1.1

Orientation

3

1.2

Points and lines

3

1.3

Surfaces

7

1.4

Solids (a look ahead)

9

1.5

Angles

10

1.6

Figure and boundary in general

12

1.7

Figures and boundaries—specific examples

13

1.8

Parallel lines—the final boundary

14

1.9

Review

15

1.10

Postulates

16

1.11

Common notions

19

1.12

Conclusion

20

Propositions and proofs—theorems and problems

21

2.1

The ancient distinctions

21

2.2

Descartes’ Géométrie

23

2.3

The context of the Géométrie

25

2.4

The geometric/mechanical distinction

26

2

vi

2.5

Descartes’ subject matter

30

2.6

Hilbert’s Foundations of Geometry

31

2.7

Hilbert’s subject matter

34

2.8

The problem of geometric subject matter

37

2.9

Figures

38

2.10

QED/QEF: figures and diagrams

40

2.11

Overview of Book I

44

2.12

Summary

47

Part II The development of methods of measurement in Euclid, Dedekind and Kronecker 3

49

The contexts of measurement

51

3.1

Introduction

51

3.2

Part and measure

53

3.3

Ratio and proportion

56

3.4

Books V and VI—propositions about proportions

58

3.5

Magnitudes and multitudes

63

3.6

Numbers and their parts

67

3.7

Commensurable and incommensurable, rational and irrational

70

Number theory in the nineteenth century

75

4.1

A shift in subject matter

75

4.2

Kronecker, Dedekind and their predecessors

76

4.3

Dedekind’s mathematical method

80

4.4

Dedekind’s struggle towards algebraic number theory. Phase I: the discovery of ramification

81

4.5

Phase 2: first introduction of fields and ideals

85

4.6

The third phase of the evolution of Dedekind’s number theory

88

4.7

The final phase of Dedekind’s evolution

92

4.8

Kronecker—changing mathematical perceptions

96

4

vii

4.9

Kronecker’s views on ‘number’

97

4.10

Kronecker’s number theory

99

4.11

A return to ramification

101

4.12

Kronecker’s ‘general theory of arithmetic quantities’

104

4.13

Conclusion

107

Appendix

109

Part III Mathematical wholes and the establishment of generality 115 in Euclid, Weil and Grothendieck Introduction

117

Types of wholes

119

5.1

André Weil and arithmetic algebraic geometry

119

5.2

Weil and the Riemann hypothesis

121

5.3

Weil and The Foundations of Algebraic Geometry

123

5.4

Grothendieck’s schemes and the Weil conjectures

128

5.5

Euclid’s approach to generality and closure

132

5.6

Parts and wholes

133

5.7

Solid figures as complex wholes

138

5.8

Constructing and measuring solids

140

5.9

Summary of Books XI—XIII

141

Generality in contemporary mathematics

143

6.1

Some contemporary problems

143

6.2

Arithmetic algebraic geometry

144

6.3

Putting the pieces together

146

6.4

Homotopy theory of categories

149

6.5

Non-abelian homological algebra

152

6.6

Generality and mathematical wholes

155

Conclusion

157

Notes

163

Index

181

5

6

Figures

2.1 2.2 2.3 2.4 2.5 2.6 2.7 5.1 5.2

Cartesian multiplication Cartesian extraction of square roots Cartesian compass Desargues’ theorem Pascal’s theorem Euclid’s Proposition 1, Book I Euclid’s Proposition 4, Book I Formation of a ‘double’ point Reducible and irreducible curves

24 24 27 36 37 41 44 124 126

Preface

Day unto day uttereth speech and night unto night declareth knowledge
Ps. XIX But the contemplative life is somehow above the level of humanity
Nic. Eth. X vii This book is the product of reflection and research undertaken over a period of twenty years under a wide variety of circumstances. It has its origins in the simple questions that David Smigelskis, Charles Wegener and Eugene Garver asked and taught me to ask at the University of Chicago two decades ago and the pleasure that accompanies the attempt to answer such questions has not diminished since. The influence of the ideas of Richard McKeon will also be evident to anyone who is familiar with them. More recently the kind and encouraging words of Stuart Shenker provided the impetus to undertake the task of combining these thoughts into a systematic whole. Over these decades there has been a steady increase in the number of analyses of works from the ‘scientific literature’ and the notion that such texts can be treated as texts has become much more widespread. Readers will surely not be surprised to find in their hands an analysis which includes not only writings of Euclid and Descartes but also more modern and indeed contemporary mathematicians as well. The mathematical literature contains many wonderful examples from a range of eras and in a variety of genres. Increased awareness of the value of this literature both within and without the mathematical community is the principal objective to which this book is devoted. Special thanks are owed to the Mathematical Institute, Angus Macintyre, Bryan Birch, Aldo and Gigi and all the others who have made my re-entry into the mathematical community so enjoyable.

Introduction

Mathematical argument has received comparatively little attention over the years compared with the voluminous literature devoted to ‘philosophy of mathematics’ and ‘foundations of mathematics’. Although all mathematics is set forth in texts which argue, explicitly or implicitly, for their own version of ‘doing mathematics’ and in spite of exhortations from all sides to read and study mathematics in the ‘original’ the general view remains that mathematical facts and mathematical subject matters exist somehow independently of the texts in which they are expressed. My purpose is to demonstrate the utility of analysing mathematical texts as texts and as arguments and to suggest the ways in which the addition of this new dimension to the existing corpus of thought about mathematics can illuminate some of the problems and conundrums facing philosophers of mathematics while raising some new issues to be addressed. At the same time ‘working mathematicians’ may be able to increase the cogency and coherence of their thought through an increased awareness of the ways and means by which it can be organized and presented. The approach put forward is not a version of ‘philosophy of mathematics’ in that it does not seek the philosophical principles that may or may not ground mathematics and its methods, nor does it attempt to codify the consequences which philosophers should derive from mathematical facts for matters which are not strictly mathematical, such as the nature of thought or truth. Furthermore, in spite of the emphasis on ‘texts’, I shall not discuss questions of style, presentation or overall ‘philosophical orientation’, at least in so far as such discussions would pertain to giving form to pre-existing content. With few models of similar analysis to refer to, the possibility of discoursing on mathematical argument as well as its utility must be demonstrated. With this dual objective in view the following essay is divided into three parts, each of which provides an example of the type of analysis being promoted as well as reflections on the relationships between this and other kinds of mathematical thought and discourse.

xi

In Part I the first book of Euclid’s Elements is carefully reviewed to provide an exegesis which takes into account the complex turns and twists of Euclid’s argument. Although many thousands of pages have been devoted to the Elements since its (unknown) date of first publication, few interpreters have sought to follow Euclid as he establishes his subject matter and shapes the argument of his treatise. Much of what will be said here is therefore at variance with established patterns of interpretation of the Elements. To put the Euclidean approach into perspective, Book I of the Elements is compared with René Descartes’ Géométrie and David Hilbert’s Grundlagen der Geometrie, two of the most significant texts on the subject of elementary geometry to have been written since Euclid. The point of comparison is the nature of the geometric subject matters which are established by each author. In Part II the books of the Elements treating ratio and number (Books V–X) are examined. In these sections of his treatise, which are considerably less elementary than Book I, Euclid sets the material he has previously developed into new contexts through definitions of ‘magnitude’ and ‘number’. These are topics which have been of great concern to philosophers of mathematics since the nineteenth century when various programmes to provide a ‘foundation’ for mathematics were put forward. The methods that mathematicians have employed to characterize number systems and produce new types of measure can be directly contrasted with Euclid’s use of contexts which neither describe nor construct material but provide new ways of exploring previously developed topics. The contrast is brought out most forcefully by examining the work of two mathematicians from the second half of the nineteenth century, Richard Dedekind and Leopold Kronecker, as they developed the foundations of modern number theory. Because these two mathematicians were (and are) considered to represent two extremes of mathematical thought and method, it is particularly illuminating to see how similar their approaches are when contrasted with Euclid’s methods. In Part III the closing books of the Elements (Books XI–XIII) dealing with solid figures are analysed. Euclid brings his argument to a close by finding a type of ‘completeness’ which arises from the subject matter itself. In doing so he does not foreclose the posing of additional questions or the postulating of new areas for research, but rather demonstrates how the particular subject matter and method which he has adopted form a complete system and provide their own sense of closure. These are matters which have been of concern to contemporary mathematicians as they have sought to formulate and reformulate languages in which to express results of higher and higher generality. While the issues are of importance to working mathematicians (and

xii

have even been the cause of polemic within the mathematical community) they have drawn relatively little attention from philosophers and writers on mathematics. The work of André Weil and Alexandre Grothendieck in reformulating algebraic geometry is a case in point. Weil’s ‘universal domains’ and Grothendieck’s ‘categories’ and ‘universes’ are attempts to encompass the broadest possible range of objects and results in a single system. The issues which arise are not simply those of ‘technical mathematics’ (although their discussion requires more mathematical background than in Parts I and II) and comparison with Euclid brings out the common problematic. In a brief concluding chapter a few of the philosophical issues brought out in the analysis will be reviewed and commented upon. The method adopted here is ‘inductive’ in nature, moving from the analysis of particular examples to broader views of mathematical argument. This is in keeping with an enterprise which seeks to explore the general nature of mathematical argument in the particular arguments of mathematicians. None the less, it may be useful to compare the conclusions adumbrated by the examples analysed here with some of the issues arising in contemporary reflection on mathematics. Euclid’s text forms the ‘backbone’ of the analysis not because of its relatively elementary nature or the extent of its influence but rather because it demonstrates a wider variety of approaches to mathematical argument and a higher degree of awareness of the issues involved than any other single text in the history of mathematics. This is not to claim a superior position for it but merely to indicate the nature of the utility that a close reading of this text can have for all mathematicians and friends of mathematics. However, what follows can by no means be considered a full-scale interpretation of the Elements. Many critical issues in the text have been ignored as their discussion would not fit in with the overall objective of the argument. A more comprehensive review of Euclid’s work and related matters in Greek science and philosophy must await another occasion.

Part I The subject matter of geometry in Euclid, Descartes and Hilbert

2

1 The opening of the Elements

1.1 Orientation A vast amount of commentary has accumulated since ancient times around the Definitions, Postulates and Common Notions with which Euclid commences the Elements. In certain cases (one thinks immediately of the notorious fifth or ‘Parallel’ Postulate) much of the subsequent history of mathematics can be viewed as one extended commentary on this text! For the English reader, Heath’s edition of the Elements surveys much of this commentary up to its publication date (2nd edn, 1925) and has the benefit of being easily available.1 The analysis conducted below does not require an extensive familiarity with this material as it refers to the lines of commentary that have been generally pursued only to clarify the points at which it diverges from the various traditional approaches. At certain points, however, the analysis does turn on specific words or phrases in the Greek original. Where this is the case the appropriate philological or linguistic background is provided. The reader may find the nature of the commentary more in line with the methods which might be more typically employed in the analysis of a poem or other ‘literary’ texts rather than in the examination of a mathematical treatise. The feeling of disorientation that results is likely to be productive. To facilitate the initial stage of the analysis the content of the first seven Definitions of the Elements (together with Definitions 1 and 2 of Book XI) are laid out schematically in Table 1.1. 1.2 Points and lines Euclid’s Definition 12, ‘A point is that which has no part’, frames the entire sequence of Definitions and provides a ‘principle’ in the sense of ‘a beginning’.3 Euclid does not have reference here either to spatial

4 FIGURES OF THOUGHT

Table1.1

location or to ‘quantity’ as such. It will be seen in Part II that quantities are derived terms in the Elements, arising from specific ways to analyse specific things. Instead of these more frequently found approaches, ) as he will again at other Euclid employs the term ‘part’ (Greek turning points in his argument, and a negative grammatical construction (the only other Definition to use negation in Book I being the last Definition, that of parallel lines, which shares a framing function with this Definition) to indicate a limit or an extremity. The limit in question is not a limit of magnitude or position, rather it is a limit of intelligibility, an extreme of discourse. In this opening phase of the Elements ‘parts’ are things in terms of which other things can be defined, discussed and understood. That which has no part merely has existence—completely undifferentiated existence. One does not have to construct or imagine such things; anything can be considered to be a point if it is considered without parts.4 All such things are undifferentiated: one cannot speak of singular and plural, of here and there, or indeed of any other type of qualification.5 Hence this definition marks the limit of that which can be discussed or analysed. Everything else which will enter into the argument of the Elements will have ‘parts’ of some kind, and determining the nature of these parts and their relationships to each other and to the wholes which they constitute is the method of Euclid’s science. Euclid’s choice of this rather strange formulation for defining ‘point’ will be seen to be reflective of his entire approach. It should be superfluous to add that Euclid is not here attempting to describe (well or poorly) the visual appearance of a point, nor his, our or the ideal mathematician’s intuition or perception of what a point is. Any reading of Euclid along these lines (Heath’s notes contain a number of examples) omits all of the aspects of this definition which differentiate it from other similar definitions used by mathematicians ancient and modern. While it may have been traditional then as now

THE OPENING OF THE ELEMENTS 5

to begin geometry texts with the definition of a point, whatever the tradition demanded Euclid has provided a definition which provides its own justification for commencing his argument. It is sufficient to note that he begins his argument with that which indicates the beginning of discourse; to refer to extraneous reasons for the formulation or positioning of this definition is unnecessary. The orientation provided by Definition 1 clarifies the sequence which follows. Unlike a point, a line does have parts, one part in particular which is labelled length.6 By referring to something with one part Euclid shows clearly that he does not have in mind a sense of ‘part’ which has to do with division or with ‘sub-objects’. Parts are the ways in which things may be known or described. As far as their definition is concerned, lines are things about which one can know only one thing, their length. For convenience this type of definition will be referred to as definition by specification of a measureable (in this case length). It is inappropriate to think of length in this context in geometric, metric or measure theoretic terms. All of these approaches require previous specification of some type of measure or some kind of line. Definition 2 merely states that lines (a) are distinguished from points by having parts, (b) are distinguished from other geometric things by having only one part and (c) can be compared amongst themselves by this part, their length. To repeat, anything can be considered to be a line, provided it is considered as having only one characteristic or term of comparison. Clearly there is no distinguishing (at this stage in the argument) between ‘two’ lines of the ‘same’ length. The grammar of singular and plural is limited precisely to distinctions of length. It is also inappropriate to think of this definition as embodying some (primitive) type of ‘dimensional’ analysis. As can be seen from Table 1.1, each item in the sequence ‘point’, ‘line’, ‘surface’, ‘solid’ is defined in a (slightly) different manner. The distinguishing characteristic of dimensional definitions is that the things to be defined are defined in the same way except for the number of dimensions involved. Euclid’s sequence by contrast has a form or shape and each term occupies the place it does for a particular reason. It therefore cannot have a ‘dimensional’ character, even a primitive one. The Elements opens with the definition of point because point is the extreme of discourse. It continues with lines and surfaces because these have parts through which they can be known. Eventually we will see why it terminates with solids. Now if lines are those things which differ from one another in precisely one way how does one actually go about making comparisons between lines? The answer is provided by Definition 3 which, as far as most commentators are concerned, does not define anything at all. This Definition states that points are the extremities7 of lines, i.e. the

6 FIGURES OF THOUGHT

comparisons of lines as lengths is effected by means of points as their extremities. One might say therefore that Definition 3 defines delimited lines by specifying the means by which this delimitation takes place. There are two aspects to this: • points can function in this way without further specification just because there is nothing further to say about a point other than that it is a point; • by functioning as the extremities of lines, points acquire a further characteristic which allows them to be differentiated, i.e. one can now speak of singular and plural, point and points. Once again it should be clear that Euclid is not attempting to describe how lines and points ‘look’, nor is he asserting that lines are ‘made up of points’ or foreshadowing some notion of ‘incidence’. The purpose of these Definitions is to define terms and establish a subject matter, not to describe already known or existing things. Definition 4 is of ‘straight lines’ and is, on most readings of the Elements, virtually unintelligible. The approach proposed here, however, suggests a simple and clear interpretation. As points define the delimiting of lines, straight lines are precisely those for which no additional specification of the relationship between points-as-extremities and lines-as-delimited is either necessary or possible. For straight lines this relationship is always the same. No metric or measure theoretic specification is implied here. The delimitation happens in the same manner throughout the line, but nothing is said about what this manner may be.8 Definitions 2, 3 and 4 taken together provide the paradigm in Euclid’s mathematics of a ‘measured thing’: • that which is measured (in this case lines) is defined in terms of a measurable (in this case length); • the measurable is further determined by specifying how the measuring or delimitation is to be performed (in this case by pointsas-extremities) and this specification requires nothing further (there is nothing further to say about points); • a special type or kind of measured thing (in this case straight lines) is defined by requiring that the delimitation of the measurable does not vary with what is delimited and that no further specification of this relationship is necessary. This must be understood as a formal procedure which moves mechanically step-by-step to define and determine the measured thing. Once the starting point (!) is adopted there is no opportunity for

THE OPENING OF THE ELEMENTS 7

variation or deviation. This is Euclid’s approach to ‘rigour’ in mathematics and failure to understand his method leads to the view that some ‘selection’ process based on tradition or philosophical inclination led Euclid to pick the particular formulations and sequence of definitions that we find in the Elements.9 The importance of a clear understanding of the nature of Euclid’s argument goes beyond an appreciation of the clarity and cogency of his thinking; it is fundamental to understanding his subject matter. As has been shown, points, lines and straight lines can be completely and satisfactorily related to one another through the definitions themselves. Because of this they do not pose by themselves interesting mathematical problems for Euclid and do not constitute the subject matter of the Elements. None of this can be said for surfaces. 1.3 Surfaces Definition 5 of surfaces mimics Definition 2 of lines with two parts, length and width, instead of length alone. The remarks above on the definition of line carry over with no substantial modifications. Clearly, once again, spatial intuition is not at issue. Nor should it be thought that a specific combination of length and breadth in a notion of area is referred to. The two measurables, length and breadth, exist independently and, at least at this stage of the argument, there is no way to put them together. Definition 6 mimics Definition 3 in specifying lines as the extremities of surfaces as points are of lines. However, the grammar of the plural ‘lines’ differs from that of points. As noted above, it was precisely in functioning as the extremities of lines that any distinctions between points could be drawn. From their definition alone there are no characteristics which would permit either numerical or generic distinctions among them. But lines do have a variety of ways in which the singular/plural distinction can be applied: • lines can differ by length, • some lines can be distinguished as straight lines, • lines can act as the extremities of surfaces (allowing distinctions among lines of the same length). Now in delimiting surfaces, lines act as extremities in that they limit the double comparison of length and breadth to one term. This does not imply or involve a ‘picture’ of lines ‘cutting’ surfaces but simply a formal relating of definitions to each other. There is no reason in this

8 FIGURES OF THOUGHT

context to limit the lines which can function as the extremities of surfaces to straight lines. The twist comes in Definition 7, of plane surfaces, which mimics the definition of straight lines but which requires that plane surfaces be those which lie evenly with the straight lines upon them. In other words the functioning of lines-as-extremities can be stated in terms of lines in general whereas the relationship between lines-as-extremities and surfaces-as-delimited is stated in terms of straight lines only. Those who take the formulation and sequence of these definitions as principally governed by external rationales may be tempted to overlook this difference in wording as symptomatic of a lack of rigour or formalization in Greek mathematics. The lack of commentary on this point historically in any event might lead one to think it of little consequence.10 To the contrary, the subject matter of the Elements arises precisely from the complex interplay between Definitions 6 and 7. The definition of straight line was noted above as paradigmatic of Euclid’s approach to defining measured things. The definitions in that case were completely sufficient to provide the required determinations. In the case of surfaces this is not so. While no a priori limitation can be set upon the lines which function as the extremities of surfaces, to achieve the complete specification of the relationship between lines-as-extremities and surfaces-as-delimited it is necessary to prescribe that the delimitation by lines happen in the same manner throughout the surface with respect to straight lines. Without this requirement the relationship would not be completely specified, i.e. while one would know that the delimiting lines operated evenly as extremities of the surface one would have no specification of the measurement of the lines themselves. Further data would therefore be required to obtain a complete determination of the thing under definition. This was not the case in the otherwise similar definitions regarding lines in that there are no ‘kinds’ of points, hence stating that points functioned as the extremities of lines was sufficient to define the relationship between points-as-extremities and lines-as-delimited. In contrast to the complete specification provided by the definitions of lines, the definitions of surfaces leave an open area to be explored. This open area is occupied by the figures of Books I–IV of the Elements, each of which represents a particular instance relating a delimitation of a surface to the surface which is delimited. The formal, mechanical nature of Euclid’s argument in the first few definitions has led to the discovery of this problematic area. Only if the definitions are viewed as proceeding according to a rigorous method can the reader make this discovery as well. Failure to appreciate Euclid’s argument has led to a concomitant failure to see how this

THE OPENING OF THE ELEMENTS 9

subject matter is discovered and indeed a failure to understand the nature of the subject matter at all. 1.4 Solids (a look ahead) The reader who is sceptical that the seemingly small matters of ‘wording’ discussed here can have any real bearing on the nature of Euclid’s enterprise may find it useful to glance ahead to the outcome of the series of Definitions (see Table 1.1) in the definitions of solids in Book XI. Definitions 1 and 2 of Book XI mimic the definitions of line and surface and their extremities but fail to provide an equivalent for the definitions of straight line or plane surface. What accounts for the lack of such a definition, apart from a failure of intuition of ‘higher dimensional’ spaces in which solids might be ‘flat’ (as contemporary geometry would have it)? What after all is there to prevent Euclid from including a transcription of Definition 7 which might run ‘A “flat” solid is one which lies evenly with the plane surfaces upon itself’? The answer is that for Euclid such a definition would be purely formal. It has just been shown that the relationship between the extremities of surfaces and surfaces cannot be stated in general. It has also been intimated that this relationship can only be explored through a set of instances using the concept of ‘figure’. Now if surfaces themselves are only known through particular instances it makes no sense to attempt to relate solids to surfaces generally, i.e. through definition. While the proposed definition looks plausible it is in fact meaningless and its omission tends to confirm the line of argument put forward here. A fuller discussion of these issues, however, will be postponed to Part III. The discovery of the open area between Definitions 6 and 7 forces Euclid to terminate the sequence of definitions which involve things which are constituted by measurables. The angles and figures which will be defined in Definitions 8–23 are not given definition by being constituted by a measurable such as length or breadth. Instead, as we will see, they are constituted by parts (i.e. they are made up by points, lines and surfaces and their definition is in terms of this constitution) in a different sense and their characteristics depend on these parts. In Book XI solids will be defined in terms of the measurables length, breadth and width. Between Definition 7 of Book I and Definition 1 of Book XI Euclid introduces no new geometric material!

10 FIGURES OF THOUGHT

1.5 Angles Returning to Book I, following Definition 7 Euclid takes a new tack and halts the sequence of definitions which have been providing ‘material’ for his subject matter. The remaining definitions operate by placing the material already defined into new contexts in which it can function in ways which go beyond the definitions which introduced the material in the first place. An analytical layout of the complete series of definitions of Book I is shown in Table 1.2. Having established a problematic area in the relationship between lines-as-boundaries-of-surfaces and surfaces-as-bounded-by-lines Euclid begins to explore this area by defining contexts in which the material defined up to this point can function to bound or contain surfaces. The first such context is that of angle, in which lines, which are defined in terms of length, function to ‘contain’ an angle. Definitions 8–12 concern themselves with this. Euclid defines angles as the ‘inclination’ of one line to another and specifies that the two lines must not lie in one straight line. This is not meant as a vague description of what an angle ‘looks like’, nor is it a definition with roots in Greek mathematical tradition; indeed it would appear that Euclid’s definition is in fact a departure from tradition.11 Furthermore, this definition contrasts quite curiously with the Definitions he gives in Book XI where it is not angle that is defined in terms of inclination but rather the reverse.12 A modern mathematician might find the latter procedure somewhat more familiar since current practice usually consists of specifying the measurement prior to the thing measured. From this modern point of view it is even more curious to see Euclid in Book I defining right (respectively acute and obtuse) angles through phrases such as ‘angles equal to’ (respectively less than and greater than) other angles without having previously defined the measurement of angles. The clue to resolving these problems of interpretation can be found by examining the role of the term ‘angle’ in Euclid’s argument. Just as points and lines acquire characteristics beyond those set out in their definitions when functioning as the boundaries of lines and surfaces, so through ‘containing’ angles do lines in general and straight lines in particular acquire further characteristics as well. But whereas point, line and surface can be defined in terms of parts which are measurables and thus provide definitions which are independent of the way in which they are bounded, the parts of angles are lines. In other words, the lines which ‘contain’ an angle are both its ‘extremities’ and its defining parts. Angles are simply a context13 in which two lines can function to define and delimit something which is not a line. This is to

THE OPENING OF THE ELEMENTS 11

Table 1.2

be contrasted with points, lines, surfaces (and solids) which are given definitions ‘as something’, i.e. are given definitions through a measurable, as well as being given definition ‘as something limited or bounded’ (Definitions 2 and 3, 5 and 6). In particular, such a context can be in a ‘plane’ and thus two lines can operate to define and delimit something in a plane. Angle is thus a specific case of the problematic area in the relationship between Definitions 6 and 7 and provides a particular example of the joint facts: lines-as-delimiting-surfaces and surfaces-as-bounded-by-lines. Given that Euclid’s objective is the exploration of this problematic area, what is important in his definitions is not the process of measurement but the constitution of the thing defined. To define a measurement of angle separate from the definition would run against this line of reasoning. As noted above Euclid does not define angles as a measurable quantity such as length, nor does he state how to measure angles and then go on to specify that two angles are equal if they have the same measurement. Instead he defines what a plane angle is through its parts (which are now14 not some type of quantity or thing measured but the things which ‘contain it’) and then goes on to set up circumstances in which angles may be compared and the consequences of these comparisons once drawn. Both the process of measurement and what is measured arise in the geometry and neither is specified in advance or through the definition. This type of argument will be encountered at many points in the discussion to follow and the reader will have an opportunity to become more familiar with this aspect of Euclid’s mathematics which deviates widely from modern mathematical practice. Euclid’s procedure is reinforced by the manner in which right, acute and obtuse angles are defined. The definition of right angles specifies a context (namely adjacent angles) in which two angles appear and can be compared. At this stage of the argument of course we have no way

12 FIGURES OF THOUGHT

of knowing how such a comparison is to be performed. Nonetheless Euclid can define right angles as equal adjacent angles because, in his approach, the notion of equality is prior to that of measurement. One cannot measure something unless one knows what it means for that which is measured to be equal to that which is measuring it. Equality is not a possible outcome of making measurements, it is rather one of the concepts which underlie the possibility of measurement. Similarly with greater than and less than. By viewing lines in the context of angles as functioning as their parts, Euclid provides himself with the first instance of ‘boundings’ of plane surfaces. However, angles are severely limited as a context in that they involve only two lines. To provide himself with a broader set of contexts in which more than two lines can function as boundaries Euclid ‘generalizes’ the concept of extremity in such a way that it can be turned into a series of specific instances of boundaries and things bounded. 1.6 Figure and boundary in general Of all the peculiarities in the set of Definitions in Book I, the presence of Definitions 13 and 14 in the middle of the sequence is perhaps the most perplexing. Why does Euclid find it necessary to define ‘boundary’ and ‘figure’ (in a rather circular way at that) at this point? If the mathematics involves figures such as circles and triangles why not simply go on to define them directly? Conversely, if figures in general are of interest why study the individual cases? The answer lies in the problematic raised by the interaction of Definitions 6 and 7. Euclid’s approach to the problem of relating the boundaries of surfaces to what is bounded is not to attempt a general theory of boundaries and surfaces (in spite of the appearance given by Definitions 13 and 14) such as a modern mathematician might be tempted to undertake. Rather he provides a context in which previously defined and determined materials can be placed so as to arrive at examples or instances of boundings. This context is provided by the term ‘figure’ (Greek In its most general sense for Euclid, a boundary is that which is an extremity of something (as points are of lines, lines of surfaces and lines of angles; Euclid introduces no new types of extremities until Book XI). which, curiously, is the word for The Greek for ‘boundary’ is ‘definition’ as well. Definition 13, which defines boundary, is therefore a definition of ‘definition’ and it states in effect that the things which can be defined are those things which have extremities or limits. This has been foreshadowed in Definitions 1–12 where it has been seen that points are the limits of intelligibility, lines are delimited, and thus

THE OPENING OF THE ELEMENTS 13

known, by points and surfaces and angles are delimited and ‘defined’ by lines. The general definition of boundary has as its correlative the definition of ‘figure’, as that which is contained by, i.e. defined by, boundaries (as lines are contained by points, surfaces and angles by lines). The meaning of ‘contained by’ here is not that of topology or set theory but rather the very direct sense in which an angle is contained by the lines which are its parts and which make it up. Boundary and figure are correlative terms which are tied together as ‘half/double’, one term implying the other.15 As boundary is the most general instance of limit or extremity, so figure is the most general instance of that which has a boundary or limit, the most general instance of that which has a definition. Angles differ from points, lines and surfaces in that the lines which limit them are also the parts which make them up. This correlation (one is tempted to say ‘covariance’) is crucial in stating and proving theorems since the parts and what they make up reciprocally determine one another. This is not the case with points, lines and surfaces—the only way to state theorems about ‘length’ or ‘breadth’ would be to assume some metric characteristics for the objects under study and then to spin out the consequences of these assumptions. Such a form of argument was known to Euclid and was in fact used by him in his treatise the Data, which is much less well known than the Elements.16 The key to the argument in the Elements, however, is the use of figures to provide instances of lines bounding surfaces and to treat the figures thus defined in terms of the parts which make them up (points, lines and angles) and not in terms of the parts of the parts (length and breadth). By putting points, lines and angles together in, say, a triangle, Euclid creates an example of a complex boundary containing a figure which can be studied because the parts of the boundary are already defined things. Putting the parts together without the context of figure and boundary would mean that further discussion would have to take place in terms of the parts themselves (e.g. length or length and breadth). By making the parts of figures correlative to figures Euclid sets up the possibility of a mathematical argument which proceeds by discussing the whole in terms of the parts without having to delve into the constitution of the parts.17 1.7 Figures and boundaries—specific examples Following the general definitions of boundary and figure Euclid gives a series of definitions (15–22) which proceed in terms of the complexity of the boundary of the figure, i.e. the number of lines which constitute

14 FIGURES OF THOUGHT

the boundary. All of the definitions operate by defining the boundary of the figure following the equation of boundary with definition in Definition 13. In the case of a circle, the boundary is a single line without other parts and is defined in terms of its relationship to the centre point of the circle. In connection with the equal radii which form the basis for the definition, Euclid follows the procedure previously noted in the discussion of the definition of right angle whereby equality operates as a basic concept of measurement and precedes all specification of the measurement of the thing in question (in this case lines). In other words, circles are contexts in which the equality of lines can be determined. In the case of rectilinear figures, these are defined in terms of the parts of their boundaries and the relationship of these parts to each other. The definitions proceed from the circle which has one line as its boundary, to the semicircle which has two lines, to the triangle which has three and so on. These definitions do not require further elaboration here. 1.8 Parallel lines—the final boundary The last definition breaks the sequence of ‘figures’ and provides a final ‘limit’ to the definitions in a manner reminiscent of the initial ‘limit’ provided by Definition 1. The two definitions share a negative grammatical construction and both indicate extremes of discourse and intelligibility. Parallel lines are the final context in which the materials of Definitions 1–7 are set. The context is defined by negation and is in certain respects the ‘contrary’ of angle. As will be seen, the conventional view of the function of parallel lines in Euclidean geometry in specifying the curvature of space is not an appropriate reading of the Elements. The rather convoluted wording of Definition 23 is an indication of the intent behind it. By defining parallel lines as lines in a plane18 which do not meet regardless of how far they are extended, Euclid is indicating that there are no a priori limitations on the extent of a plane. By Definition 13 it is clear that Definitions are of limits—Definition 23 operates through negation of limits. For this reason parallel lines do not contain anything. In a metaphorical sense it might be said that Definition 23 specifies that plane surfaces are ‘big enough’ to contain parallel lines or that parallel lines are the measure of the (potential) extent of plane surfaces. The two reciprocally delimit each other by specifying a lack of limitation or absence of boundary.

THE OPENING OF THE ELEMENTS 15

Definitions 1–7 : Definitions 8–23:

parts=terms of definition limits=measures of parts limits=terms of definition parts=measures of limits

This definition stands at the limit of intelligibility because it succeeds in defining without reference to boundary. There are no terms to discuss things without parts; there are no definable things beyond that which has no boundary. 1.9 Review The first seven definitions of Book I provide materials which are then placed in specific contexts in the remaining definitions. The definitions of this material indicate that definition as a procedure is insufficient to provide its complete determination, hence the sequence of definitions through measurables breaks off and is complemented by definitions of things through their boundaries and the parts of these boundaries. The above discussion can be summarized succinctly by relating the terms part, limit and definition in two rather paradoxical ways: Euclid’s procedure is first to locate a problematic area which indicates the presence of a possible subject matter and then to provide the materials with which to explore it. His definitions provide a subject matter which has structure and coherence but retains its problematic nature. In spite of their detailed and elaborate structure, the definitions themselves in no way ‘prejudice’ or predetermine the conclusions of the scientific investigation to follow. The argument Euclid employs is as rigorous as that of any formalized mathematical language and it is precisely its mechanical ‘step-by-step’ nature which makes possible the discovery of a potential subject matter in the relationship between lines-as-boundaries and surfaces-as-bounded. Figure then provides a general context (similar to that of angle but more flexible in the number and kinds of parts it can accommodate) in which definition can take place and the parts of figures, i.e. the terms in which figures are defined, become directly related to the figures themselves. The various figures then defined are specific instances of boundaries and boundeds, each of which can be studied because it is a construct of parts. The nature of the subject matter drives the argument. Euclid is not free to include or exclude elements, nor to alter or reformulate definitions to satisfy the dictates of tradition, philosophical orientation or pedagogical requirements.

16 FIGURES OF THOUGHT

These definitions are in no way based on intuition nor do they describe existing objects. Rather the entire thrust is to establish a subject matter, i.e. an area where questions arise and can be answered. Prior to attacking the subject matter however, there are additional specifications and determinations which are required. These are set forth in the Postulates and Common Notions, both of which refer back to the subject matter of the Definitions. 1.10 Postulates As it is in connection with the Postulates that most interpretative attention has been focused on the Elements by mathematicians it will be useful to review certain preliminary issues here quite carefully. and the phrase ‘let it be The word ‘postulate’ in Greek is postulated’ is a translation of the passive perfect imperative verb form The stem of this verb is the passive voice carrying the sense of being the passive perfect ‘to be required or requested’, and, imperative may best be translated as ‘let it already have been required’. Each of the postulates or requirements relates back to one or more of the definitions and specifies that the data of the definition are not only necessary but also sufficient for the determination of the thing in question. The perfect tense indicates that the specification or determination has ‘already happened’ and that there is no temporal or logical sequence into which it can be analysed or broken down. Specific examples will make this clearer. Postulate 1 requires that given two points there be a straight line between them, i.e. two points immediately determine a straight line for which they are the extremities. Definition 3 defines the extremities of lines to be points (thus in reality defining delimited lines) and Definition 4 specifies that straight lines are those in which this delimitation operates in the same manner throughout. No additional data is required to understand or specify how this delimiting takes place. Postulate 1 reverses Definition 3 and requires that points be sufficient (as well as necessary) to determine a delimited line. This postulate is not part of a set of artificial constraints on mathematical constructions such as are set out in so-called straight edge and compass constructions, except perhaps in the most attenuated sense. The postulate’s function is to remove the possibility that any data other than its extremities, the minimum data required, would or could be necessary to determine a delimited line. Conversely, without the postulate there could conceivably be the need for such data and Euclid would have to provide it in some other fashion (and this is in fact the

THE OPENING OF THE ELEMENTS 17

manner in which most geometers, ancient and modern, have proceeded). Note that the determination of the line given the points is immediate. It does not take place through the agency of some means of construction or through some logical consequence. There is no ‘middle term’ between the points and the line between them. The verbal form of postulation suggests that we view the line as already there once the points are given. This contrasts with the nature of mathematical construction in the Elements as will be seen when Proposition 1 is analysed. Postulate 2 states that there are no restrictions on the extension of a straight line. It is a ‘self-determining length’. Whereas Postulate 1 reverses Definition 3 by stating that points are sufficient to determine a delimited line, Postulate 2 reverses Definition 4 by requiring that a straight line have no limitations a priori. Points determine the delimitation of a straight line in the same manner throughout and no point is such (or can be differentiated in such a way) that the line cannot be extended beyond it. With Postulates 1 and 2 straight lines become self-determining. The data required for their definition and delimitation are sufficient for their determination. Postulate 3 then does the same for circles. The parts which are necessary to define a circle—centre and radius—also suffice for its determination. This Postulate is no more a part of a ‘ruler and compass’ programme than are Postulates 1 and 2. Postulate 4 has attracted the least attention from commentators but provides in fact the clearest example of Euclid’s method. Right angles are defined in Definition 10 as equal adjacent angles produced when one straight line is set upon another. The definition itself is insufficient to rule out the possibility that different sets of right angles might not be equal to one another. In such a case additional ‘external’ specifications or determinations would be required to distinguish and relate the various sets of ‘right angles’. Postulate 4 removes such a possibility by making the elements of their definition sufficient for the determination of right angles. Postulates 1–4 cover points, lines, angles and circles, thus only surfaces, rectilinear figures and parallel lines remain to be addressed. Postulate 5 deals with all of these. Recall that the definition of parallel lines was phrased in the negative; this definition is a peculiar type of limit as it defines something which is not delimited. Postulate 5 is therefore not about parallel lines as such in that it does not provide a sufficient condition for lines to be parallel in the way in which Postulates 1–4 provide determinations of straight lines, angles and circles. Instead Postulate 5 provides a determination of non-parallel lines (and therefore, implicitly, of triangles). This Postulate depends on

18 FIGURES OF THOUGHT

Postulate 2 (to be able to extend the lines) and Postulate 4 (providing right angles as a stable measure). The further determination that Postulate 5 provides is that the non-parallelism of the lines as extended is dependent only on the angles formed by the transverse line. That these angles are measured against right angles is due to the fact that right angles are the only stable measures available. Additional ‘external’ determinations would be required if the standards of measurement chosen were not right angles. From this point of view, Postulate 5 is not a selection of a ‘type’ of geometry (one among many), an attempt to prescribe spatial intuition as fundamental or even a ‘self-evident’ truth. This Postulate, like the others, provides a kind of self-determination within the material given by the definitions so that the parts necessary to define a given element also suffice to determine it. More particularly, without Postulate 5, data would be required to specify the relationships between lines and angles in figures apart from the lines and angles themselves. The long history of commentary and analysis of this Postulate, by treating it as a proposition instead of a postulate (one of the natural consequences of the usual understanding of ‘axiomatization’ as will be seen when Hilbert’s geometry is discussed) has overlooked this simple functioning of Postulate 5 within the argument of the Elements. The determination of non-parallel lines in Postulate 5 bears an important relation to Definition 23. The negative definition of parallel lines indicates the lack of bounds to the size of planes or plane figures. Postulate 5 provides a determination of the existence of triangles based solely on the data of transverse lines and angles without limit as to ‘size’. Together, Definition 23 and Postulate 5 eliminate the possibility that external data will be required to decide on the possible limits to figures. Euclid’s material is completely self-determining and is not subject to constraints outside of its own definition. With the five postulates all of the elements supplied by the definitions have been provided with a minimum level of self-determination and self-sufficiency. Without the postulates, as noted above, there might be need in certain cases for external specification of, for example, right angles. With them, such possibilities are foreclosed. Furthermore there can be no other postulates since there are no other elements from the definitions to address. The subject matter drives the argument. Euclid is not free to select a set of postulates according to philosophical predisposition, pedagogical efficiency or a subjective sense of beauty in mathematics.

THE OPENING OF THE ELEMENTS 19

1.11 Common notions While the Postulates relate to specific items within the series of definitions and provide for their determination, the Common Notions, as their name implies, apply across the entire collection of defined things.19 The Common Notions do not deal with the measurement of specific defined things, but with measured things in general, providing the ability to relate measured things to each other and thus to create propositions. In a sense the Common Notions provide the basic syntactical connections for Euclid’s mathematical sentences.20 Common Notions 1–4 provide a syntax for the use of the word ‘equality’ which Euclid never defines as such. As noted above, Euclid takes ‘equality’ to be a fundamental concept or principle of measurement in contrast to the modern preference for looking at ‘what is equated’. Hence he provides no specification of how the operations of addition, subtraction or superposition to which the Common Notions refer are to be defined. These will vary with the things to be equated. The role of the Common Notions is to spell out the types of circumstances or situations (contexts) in which equality and inequality can be found. In particular, with Common Notion 4, Euclid provides a specific type of context in which equality can be identified or located. Although the text speaks of ‘superposition’, there is no concept of ‘rigid motions’ or any of the other paraphernalia of modern geometry at work here. All of the modern approaches assume the figures and their properties as already known and go about providing an efficient axiomatization of these properties. Euclid’s point here is simply that if the superposition of two things can be established (note the generality—points, lines, surfaces or any other measured things may be superimposed) then the things so superimposed are equal. At this stage nothing is said either of the means by which superimposition is to be effected or of the measurables that may be thus related. This point will be returned to in reviewing Proposition 4 of Book I. It may be useful to compare Common Notion 4 with Common Notion 5 which identifies part and whole as a context in which inequality and ‘greater than’ or ‘less than’ can be established. The two Common Notions operate on the same level of generality as to the things that may be compared and hence neither provides a specification of how the comparison is to be performed, i.e. of how the relevant context (superposition or part/whole) is to be demonstrated.

20 FIGURES OF THOUGHT

1.12 Conclusion Far from exhausting the consequences and implications of the opening pages of the Elements the above remarks merely provide an outline of a reading of the text which attempts to account in a positive way for its quirks and turns. Euclid’s approach to establishing a subject matter, providing the elements of it with sufficient self-determination and providing as well the ability for statements about these elements to be connected in propositions should now be clear. Its value will be best appreciated in seeing it in action. Rather than proceeding directly to Book I however, it will be helpful at this point to bring in the work of Descartes and Hilbert whose very different approaches will serve to set off and highlight the distinctive characteristics of Euclid’s mathematics.

2 Propositions and proofs—theorems and problems

2.1 The ancient distinctions The Definitions, Propositions and Common Notions provide a constitution for Euclid’s subject matter but are not themselves part of it. As a consequence the qualifications of true and false which are characteristic of propositions cannot be applied to them. When we turn to Euclid’s propositions beginning with Book I we find that they fall into two types—theorems and problems in the vocabulary of ancient Greek mathematics. Simply put, a theorem is a statement about a figure which is to be demonstrated and a problem (or construction—the two words will be used interchangeably) is a request for something to be constructed given some data. While there are a few propositions in the later books of the Elements which do not fall into one of these two categories, the overwhelming majority of Euclid’s propositions are of one of these two types. The distribution of the two types of proposition, however, is quite complex. Book I is made up of an alternating series of theorems and problems (see Table 2.1) while other books may be mostly or entirely theorems (e.g. Books II and V) or problems (e.g. Book IV). This rather innocuous looking distinction carries a great deal of significance for the nature of Euclid’s mathematics.1 Proclus (410–485 AD), the neo-platonic philosopher and commentator on Euclid, recounts that from the time of Speussipus (410– 339 BC) and Menaechmus (390–337 BC) there was much philosophical dispute over whether mathematical propositions were in fact either all theorems or all problems—so that the distinction disappeared–as well as much debate on the grounds according to which the distinction could be drawn.2 This debate, while originally couched in the language of Greek science and philosophy, has had a long career and taken on many guises down the ages. The thesis proposed here is that Euclid’s subject matter is located by the interaction of theorem and problem. To see what this could mean and to clarify the implications of this view for the

22 FIGURES OF THOUGHT

Table 2.1

nature of mathematical practice it may be helpful to look at the ways in which mathematicians other than Euclid have handled these issues. The discussion of the plane geometric books of the Elements will therefore be placed in a context provided by two more recent classics of geometry, Descartes’ Géométrie and Hilbert’s Foundations of Geometry. The approach of reading mathematical texts as texts, attempting to account in a positive manner for the procedures of each, will be maintained and the purpose of the discussion is not to provide a complete analysis of either of these two mathematical classics, but rather to use certain terms, concepts and methods which can be found in them to highlight the particular nature of Euclid’s argument. The ancient disputes over theorems and problems were set in terms of the objects of mathematical science. Speussipus’ thesis can be summarized as postulating that mathematical objects are eternal and hence that there is no place in this unchanging realm for the comingto-be and passing-away that is (at least verbally) implied in the formulation of problems. Menaechmus, on the other hand, takes the opposite position that in all cases the mathematician is asked to do something, to supply something which is sought after, whether this is accomplished by showing how the thing is produced or showing that it has a particular attribute. Although locating the distinction between theorems and problems in mathematical objects has much to recommend it, a somewhat different location of this distinction, in the mathematical demonstrations, will be applied here. Descartes and Hilbert provide in a sense a ‘stereoptical’ view of the matter which

PROPOSITIONS AND PROOFS—THEOREMS AND PROBLEMS 23

should be helpful in seeing how the relationships between theorems and problems affect the nature of mathematical subject matters. 2.2 Descartes’ Géométrie3 Descartes begins his book with this famous statement: All geometric problems can be easily reduced to such terms that it is only necessary from that point on to know the length of certain lines in order to construct them.4 Descartes thus begins his treatise literally by ‘having’ problems to solve as well as by assuming the concept of the length of a line. His method is to reduce any given problem, no matter how it is originally stated, to a problem in which that which is to be constructed (the unknowns) are the lengths of certain straight lines. When such lengths are known the problem is solved and the desired construction can be made. He goes on to say that since ‘Arithmetic’ is composed of only four or five operations (besides +, −, ×, ÷ he professes indifference as to whether the extraction of roots is to be considered a form of division or not): We have nothing else to do in geometry concerning the lines for which we are looking, in order to prepare them to be known, than to add others to them or subtract others from them or, having one line, which I call the unit to relate it better to numbers and which can ordinarily be taken at discretion, and having two others, to find a fourth which will be to one of the latter two as the other is to the unit, which is the same as multiplication, or to find a fourth which will be to one of the latter two as the unit is to the other, which is the same as division, or finally to find one, two or more proportionals between the unit and the other lines which is the same as taking the square, cube or higher root etc
5 Thus, from the outset, Descartes assumes these operations on numbers to be known and proceeds to show us how they can be performed on lines as well. Figures 2.1 and 2.2 show his constructions for multiplication and extraction of square roots. Descartes continues by pointing out that it is frequently not necessary to draw the lines—one can merely denote them by letters, each line by a different letter. He reminds us, however, that by expressions such as a2 and b3 he has reference only to lines, although words like square and cube taken from algebra may be used for convenience.6

24 FIGURES OF THOUGHT

Figure 2.1 Cartesian multiplication

Figure 2.2 Cartesian extraction of square roots

He then suggests that to solve a problem we consider it already solved (this is the time-honoured method of ‘analysis’) and that we give names to all of the lines in the diagram that seem ‘necessary for the construction’ indifferently as to whether the lines in question (i.e. their lengths) are known or unknown. At this point: One should review the difficulty in the order which shows most naturally the way in which the lines depend on one another until a way has been found to represent one quantity in two ways: this is called an equation because the terms of one of these ways are equal to the terms of the other. And one should find as many equations as there are unknown lines.7 To accomplish this last step he permits the use of lines whose lengths are known (but otherwise taken at discretion) to relate to unknown lines in cases where the problem is undetermined.

PROPOSITIONS AND PROOFS—THEOREMS AND PROBLEMS 25

Descartes now relates these equations to one another so as to ‘explain’ each of the unknown lines until there is only one left to ‘explain’ and it is equal to a known line or a composition of known lines. He ends with the following assertion, And we can always thus reduce all unknown quantities to a single quantity where the problem can be constructed by means of straight lines and circles or conic sections or even by some other line which may be one or more degrees composed. But I will not take it upon myself to explain this in more detail because I would thus remove from you the pleasure of learning for yourself and the utility of cultivating your mind in exercising it which is the principal counsel of this science.8 He does allow himself to suggest, however, that the student make use of all of the ‘divisions’ which the problem allows in comparing the equations with each other and remarks on certain cases in which, due to the necessity of extracting a root, a solution may not be possible. He also notes that the ancients were clearly ignorant of his method since the very order in which they state their propositions is inconsistent with his approach. To summarize, Descartes assumes known the concept of length of lines, the arithmetic operations and the ability to apply similar operations to lines by analogy.9 His method is set forth as a sequence of steps in which the lengths of various lines in a given problem are related to one another resulting in a series of equations which are then reduced to a single equation and the length constructed according to the known operations. As an ‘advertisement’ for the power of his method he exhibits a solution to a problem of Pappus’ which had previously gone unsolved. 2.3 The context of the Géométrie Descartes published his Géométrie as one of the three essays in ‘method’ appended to his famous Discourse on Method. Each of these essays was intended as an example or illustration of his ‘method’ in operation. While it is not possible here to review even briefly the ramifications and development of Descartes’ views on method, it is important for what follows to recognize that the reader of the Géométrie is not only expecting to learn about that subject but also to find in this new treatment of an old science an example of a new method. Curiously, Descartes gives few clues to help in reading the book in this way. The Géométrie therefore presents a double problem for the reader trying to

26 FIGURES OF THOUGHT

grasp its subject matter in that the book doubles as a treatment of a specific science and an instance of a general method. Fortunately, as Descartes has already pointed out, if the same thing can be expressed in two different ways we can set up an analogy10 relating the known to the unknown! Descartes’ principal problem in organizing his book is how to provide it with structure and limits. As the text itself is ‘merely’ an example of a method its subject matter might be thought to be completely indeterminate. Furthermore the specific treatment of geometry that he sets out is itself only a way to solve geometric problems – virtually any problem could, in principle, have been selected. Are we then justified in reading the text merely as a collection of examples of problem solving illustrating how Descartes’ general method may be employed in this specific science? The steps of the method outlined above give no clues to the resolution of this problem. Descartes’ procedure in the remainder of the treatise does help, however, to shed light on his notions of geometric subject matter. In fact he interrupts the constructions of Book I to interject a series of remarks in Book II before concluding his solution of Pappus’ problem in Book III. 2.4 The geometric/mechanical distinction Descartes remarks at the outset of Book II that although he accepts the ancient distinction between linear, planar and solid problems he is surprised both at the failure of the ancients to explore more highly ‘composed’ lines and at their reference to all of these lines as of ‘mechanical’ as opposed to a ‘geometrical’ nature. He proposes to classify lines in a somewhat different manner, Taking as we do ‘geometric’ to be that which is precise and exact and ‘mechanical’ to be that which is not, and considering Geometry as the science which teaches generally the measurement of bodies, we must therefore not exclude composed lines any more than simple ones provided that we can imagine them as described by a single continuous motion or by several motions which succeed each other of which the latter is completely determined by the earlier.11

There has been a good deal of debate over the exact nature of the distinction Descartes is drawing here and recently its importance in understanding the nature of Descartes’ mathematics has been

PROPOSITIONS AND PROOFS—THEOREMS AND PROBLEMS 27

Figure 2.3 Cartesian compass

emphasized by a number of commentators.12 For our purposes it is important first to note that Descartes locates this distinction in the mind’s ability to imagine certain kinds of motion. The kind of motion in question is one which is either single and continuous, such as is produced by the classic type of compass, or else is composed of a connected series of motions which determine each other sequentially, there being no gaps between these motions which would allow or require choices to be made. Descartes provides an example of this type of complex continuous motion with an instrument which has come to be known as a ‘Cartesian compass’ and which is illustrated in Figure 2.3. Descartes’ geometric/mechanical distinction operates by specifying how new geometric curves can be generated out of existing material. The complex continuous movements are generated from already known curves by means of a process that a modern mathematician might be tempted to label ‘recursive’. Each newly generated curve is related to a previously known curve in a determinate manner. For example, curves traced by the ‘Cartesian compass’ begin with a circle and increase by degrees as legs are added to the compass. Similarly, in Book II, Descartes demonstrates a general method for superimposing a new ‘continuous’ motion on top of the tracing of a previously given curve to generate curves (so-called ‘Cartesian parabolas’) of arbitrary degree. The step-by-step nature of the process allows Descartes to classify the results as follows:

28 FIGURES OF THOUGHT

When the equation rises only to the rectangle of the two intermediate quantities (or, what is the same, the square of one of them), the line is of the first and simplest class, in which only the circle, the parabola, the ellipse and the hyperbola are included. But when the equation rises to the third or fourth dimension of the two or one of the two indeterminates
it is of the second class, and when the equation rises to the fifth or sixth dimension it is of the third class and so on to infinity.13 Since the original curves on which this process is based are known and since the motions derived from these curves are determinate, Descartes is able to draw the following conclusions: Thus, regardless of the manner in which the description of a curved line is imagined [i.e. regardless of the means by which we imagine its being drawn], provided that it is among those I call geometric, one will always be able to find a single equation for determination of its points
14 and further, I know of no better way of stating it than to say that the points which we may call geometric, that is which fall under an exact and precise measurement, have of necessity some relationship to a straight line and this relationship can be expressed for all such points on a given curve by one equation.15 While Descartes originally locates the geometric/mechanical distinction in the mind’s ability to imagine certain motions, he soon reformulates it in ‘philosophical’ terms as a distinction between things which are exact and precise and things which are not so and then further translates it into a mathematical description requiring that a single (algebraic) equation govern the relationship of the points on a given geometric curve to a straight line. But we have just seen that the ability to reduce’ the equations derived in the examination of the solution of a problem to a single equation is the key step in Descartes’ ‘method’. Hence, in effect, his geometric/mechanical distinction serves to equate geometric with ‘solvable’, ‘constructible’ or ‘demonstrable’. He goes on to say: We cannot accept any lines which are like chords, that is, partly straight and partly curved because the proportion that holds between the straight and the curved is not known and I believe

PROPOSITIONS AND PROOFS—THEOREMS AND PROBLEMS 29

cannot be known by men and hence nothing can be concluded from it which is exact and precise.16 The Géométrie deals with the exact and precise and its subject matter consists exactly and precisely of those things which can be constructed and demonstrated. This is the subject matter of his text. The formulation of a geometric subject matter thus operates externally to separate the singly continuous motions from the others, the exact and precise from what is not and the geometrically demonstrable from the mechanical. It also operates internally to structure the subject matter. The restriction to geometric curves, as he defines them, allows Descartes to build up his curves by degree and complexity as discussed above. This restriction then provides two structuring principles for his argument. First, the limitation to geometric curves specifies what is required to be known about the curves in question in order to use them for solving geometric problems: And simply from knowing the relationships which hold between the points of a curved line and the points of a straight line as I have explained, it is also simple to find the relations they have with all other given lines and points [in the problem in question].17 Thus, knowing this relationship suffices for the determination of diameters, axes, centres or other special points relating to the curve and hence provides a variety of ways in which the curve can be described (imagined, drawn) so that the geometer can select the simplest. It also suffices, according to Descartes, for finding out all that can be known about the areas bounded by curves and about the angles between curves. Second, the class of the curve (as defined above) can be connected with the complexity of the problem to be solved. For example, in Book II Descartes demonstrates the relationship between the number of lines in a given Pappus-type problem and the class of the curve which traces the locus of the solutions. In Book III he states that: Yet although all of the curves that can be described by some regular motion should be received in Geometry, this is not to say that it is permissible to use the first which comes to hand in the construction of a problem. Rather, care should be taken to choose the simplest curve by which the problem can be solved. Further it should be noted that by the simplest I do not mean that which is easiest to describe nor that which most easily renders the problem solvable, but principally that curve of the lowest class which can be used to determine the quantity desired. 18

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Finally, in Book III, Descartes provides a systematic way to reduce the degrees of equations that are involved in the solution of problems so that the lowest class of curve can be selected. He concludes the text by demonstrating the general solution of problems involving equations of up to fourth degree using a curve described by the intersection of a parabola and a straight line. In so doing he notes that this class of curves comes after the conic sections in the solution of ‘that problem [i.e. the Pappus problem] whose solution demonstrates in order [emphasis mine] all of the curves which should be received into geometry’. 2.5 Descartes’ subject matter Descartes’ definition of his subject matter provides an answer to the question raised in Section 2.2. The Géométrie is not simply an example of a method or even a series of examples of applications of a method to geometric constructions. Geometry in his view deals with the precise, the demonstrable, the constructible. This limitation not only sets geometry off from other things but also provides a means for giving his mathematical arguments structure by ordering the complexity of problems and relating such orders of complexity to the curves which arise in constructing them. The geometric/mechanical distinction operates within a realm of mathematics which is ‘already given’ and which exists prior to the application of the distinction. As has been noted, Descartes takes a great deal for granted from geometry, algebra and arithmetic in getting his argument started. What is the relationship between the new subject matter of the Géométrie and these subjects on which it depends? Can all of mathematics be organized around this distinction? In particular how are algebra and arithmetic related to geometry? We must (sadly) leave these larger questions unanswered. Suffice it to say that Descartes’ geometry is a realm of problem constructions within a larger and so far undefined mathematical domain. Within the realm of geometry, however, it is clear that all demonstrations are constructions and the subject matter of this realm can be identified with ‘the geometrically constructible’. Descartes the philosopher is most famous for his ‘cogito’ in which he demonstrates the indubitable existence of his thinking self. Ultimately, as an example of ‘method’ in Descartes’ philosophy, his Géométrie will reside in the mind of the geometer who has absorbed it. Descartes never tires of insisting that he is not attempting to write a definitive treatise but wishes only, as it were, to ‘show how it goes’ and ‘how it can be done’:

PROPOSITIONS AND PROOFS—THEOREMS AND PROBLEMS 31

it is not necessary to follow the same path to construct all of the more complex cases to infinity. Because in mathematical progressions, when one has the first two or three terms, it is very easy to find the others.19 Cartesian geometry is not a systematic collection of geometric propositions or solutions to geometric problems, rather it is the methodical approach to the subject matter structured by the geometric/ mechanical distinction. It resides or is located in the mind of the geometer and is given objectivity through the operation of the method. Its existence is bound up with the indubitable existence of the thinker and the thinker’s thought. It is through this thought that constructions are performed, it is in this thought that the distinction which locates the subject matter is first identified and it is in cultivation of this thought through the exercise of geometry that the principal utility of the enterprise lies. These characteristics of Descartes’ mathematics will be seen in sharper focus when contrasted with David Hilbert’s approach which opposes it in almost every respect. 2.6 Hilbert’s Foundations of Geometry20 In common with Descartes’ Géométrie, David Hilbert’s Foundations of Geometry (referred to as GdG from the German title) is an example of a method in operation. In fact Hilbert’s famous ‘programme’ for the foundations of mathematics had its genesis in his work on this text although the full-fledged formulation of the ‘programme’ only took place two decades later. For many mathematicians, the ‘foundational’ work accomplished in GdG was far more successful than some of Hilbert’s later efforts which have had such a pervasive influence on thinking about mathematics. GdG went through seven editions in Hilbert’s lifetime (and three more after his death, with the tenth and final edition having achieved a standard position) reflecting the evolution of his thought and the progress of his programme in the wider mathematical community. Whatever doubts and controversies have surrounded the larger Hilbert programme, GdG has remained a model of the axiomatic method and its publication has been hailed as the dawn of modern ‘abstract’ mathematics. Once again, in common with Descartes, Hilbert begins his text with a‘problem’, but his ‘problem’ does not in the slightest resemble a Cartesian construction and from this point forward the reader notices

32 FIGURES OF THOUGHT

a curious set of points of contact and divergence between the two texts. Hilbert states his ‘problem’ as follows: Geometry, like arithmetic, requires only a few and simple principles for its logical development. These principles are called the axioms of geometry. The establishment of the axioms of geometry and the investigation of their relationships is a problem which has been treated in many excellent works of the mathematical literature since Euclid. This problem is equivalent to the logical analysis of our perception of space.21 The Géométrie proceeded without axioms but having problems whose solutions called for construction. Hilbert opens with a focus on axioms and with a problem which requires a logical analysis rather than a construction. His brief introduction concludes with the following: The present investigation is a new attempt to establish for geometry a set of axioms which is complete and as simple as possible and to deduce from them the most important geometric theorems in such a way that the meaning of the various groups of axioms as well as the significance of the conclusions that can be drawn from individual axioms comes to light.22 Hilbert thus begins with a clear idea of ‘what geometry is’ and even what its most important theorems are. He is out to order and organize the propositions of geometry so that their interrelationships and logical dependencies are clarified. Whereas Descartes reduced the construction of all geometric problems to the knowledge of the lengths of certain lines, Hilbert organizes the demonstrations of all geometric propositions by means of certain axioms. Although Descartes and Hilbert each have a place in their notion of geometry for both problems and propositions, these terms take on very different meanings in the work of these two men. For Hilbert, problems are not necessarily constructions but, more generally, heuristic guides to the discovery of further propositions and demonstrations. In other words, unproved theorems. This can best be seen from the introduction to his famous speech on ‘The Problems of Mathematics’ given in 1900 to the International Congress of Mathematicians in Paris where he refers to problems as a means for gauging future mathematical developments and as a guide for research.23 Of the twenty-three problems that he proposed in his speech, only two take the form of a classic construction. This quite general view of the meaning of the term ‘problem’ was not accidental but rather arose from a philosophical difficulty with the

PROPOSITIONS AND PROOFS—THEOREMS AND PROBLEMS 33

notion of problems understood as constructions which Hilbert explained on a number of occasions. In Hilbert’s view, an ax iom system acts to characterize the subject matter it grounds. Mathematics is then properly argued and presented not by showing how mathematical elements may be constructed, which, according to Hilbert, gives only a generic idea as to how things may be generated and defined, but rather by axiomatization, which gives a complete definition of the particular things in question: In this way the axiomatic method is distinguished from the constructive or genetic method of grounding a theory. Through the constructive method the objects of the theory are introduced merely as a kind or species while with the axiomatic method one deals with a specific system of things (or several such systems), and these comprise a delimited realm of subjects for the predicates so that from these two together, the statements of the theory are constructed.24 The grammatical analogy employed here is reminiscent of various aspects of the discussion of Euclid except that here it is the axioms (corresponding to Euclid’s Postulates) which provide the individual ‘subjects’ for the predicates carried by propositions. The familiar problem/proposition distinction has also shifted from the ancient discussions of ‘being’ and ‘becoming’ to a discussion of generic/ specific with the result that, in effect, problems are subsumed under propositions and the process of demonstration is equated with the process of characterizing a subject matter. With hindsight we can see that Descartes too collapsed the proposition/ problem distinction, but in the other direction in that he brought propositions under problems by equating demonstration with construction. Schematically, Descartes: Hilbert:

problem solving=construction=demonstration theorem proving=characterization=demonstration

These equations allow a simple comparison of the ways in which these two authors go about establishing their respective subject matters of geometry. In Descartes the equation of problem solving with construction is a philosophical matter deriving from his method. It is the opening position of the Géométrie. On the other hand, the equation of construction with demonstration requires the definition of a subject matter through the geometric/mechanical distinction. Similarly,

34 FIGURES OF THOUGHT

Hilbert’s equation of theorem proving with characterization of a subject matter is also a ‘philosophical’ point (deriving perhaps from Dedekind’s approach to mathematical argument as we shall see in Part II). As in Descartes, however, the establishment of the second half of the equation is an integral part of what he does within the text and locates the subject matter of the treatise. 2.7 Hilbert’s subject matter So much has been written about Hilbert’s thinking concerning the foundations of mathematics and his ‘programme’ that even a brief discussion of these topics is out of the question here. It seems to be beyond dispute, however, to say that Hilbert held that whereas the existence of a mathematical subject matter was not to be demonstrated directly through construction, it was the axiom system itself, and particularly its property of ‘consistency’, which grounded the existence of the subject matter. As noted above, the axioms give the individual subjects for the predicates of the theory, a contradiction flowing from the axioms would negate the possible existence of the subjects: But it is not enough to remove contradictions that may arise
the principal requirement of axiomatics must go further, namely to show that within each scientific realm which is axiomatized it is impossible for a contradiction to arise. This I have accomplished in my GdG.25 Conversely a set of axioms which is not self-contradictory automatically grounds an existent mathematical subject matter. To demonstrate the consistency of his set of axioms Hilbert demonstrates that they have at least one consistent model, the field of complex numbers. This is accomplished in a manner which is very reminiscent of Descartes’ procedures (if not his objectives) in that an equivalence is established between the various geometric and algebraic (or, in Descartes’ language, ‘arithmetic’) operations. Any contradictions which might flow from the axioms would have to show up in the field of complex numbers. Hilbert takes this to be impossible. It should be recalled that Hilbert’s later foundational work was based on the premise that this type of ‘relative’ consistency proof could not be utilized in proving the consistency of the axioms providing a foundation for mathematics since there would be no ‘other’ mathematical system into which the foundational system could be transformed. The procedures employed by Descartes and Hilbert are therefore similar to the extent that geometry and algebra are linked to each other,

PROPOSITIONS AND PROOFS—THEOREMS AND PROBLEMS 35

but the similarity ends at this point. Descartes ‘reduces’ the solution of geometric problems to the problem of constructing certain lines using geometric equivalents of algebraic operations. Hilbert demonstrates that an algebraic system can be interpreted as satisfying certain geometric axioms thus showing that the axioms are consistent. Yet while this process of interpretation or modelling provides Hilbert with a proof of the existence of the system he is axiomatizing, it also opens up another set of questions. If the number system that Hilbert employs is merely a model of the axiom set he has laid down, are there others? Would they be equally valid? If the answer to these questions is yes, as would seem to follow from Hilbert’s argument, what is the actual subject matter of Hilbert’s science? Is the treatise in front of us just one example among many that Hilbert could have written, each providing a different model of the axiom system? This is clearly the analogue of the question raised in connection with Descartes in Section 2.3. A closer look at the text once again provides the clue. After establishing the consistency of his axioms Hilbert goes on to demonstrate their ‘independence’, i.e. to demonstrate that no one of the axioms can be deduced as a logical consequence of any of the others. His proof of independence involves the construction of geometries— systems whose elements satisfy some but not all of the original axioms. As the total original collection was consistent, any subcollection will also be consistent so its ‘geometry’ will ‘exist’. By then constructing models of these geometries, Hilbert demonstrates that the axiom sets can be assumed to apply or not to apply independently of one another. More precisely, after a chapter in which he provides himself with more tools to link geometry and algebra, he goes on to demonstrate in a truly spectacular way that: • a theory of plane area can be derived from the axioms (but not a theory of volume); • Desargues’ theorem, which states that if two triangles are situated in a plane so that pairs of corresponding sides are parallel then the lines joining the corresponding sides pass through one and the same point or are parallel (see Figure 2.4), expresses a criterion for a ‘plane’ geometry to form part of a ‘space’ geometry26; and • Pascal’s theorem, which states that if A, B, C, and A’, B’, C’ are two sets of points on two intersecting lines and if AB’ is parallel to BC’ and AA’ is parallel to CC’ then BA’ is also parallel to CB’ (see Figure 2.5), is dependent in a very specific way on the so-called Axiom of Archimedes.27

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Figure 2.4 Desargues’ theorem

Figure 2.5 Pascal’s theorem

None of these statements can be given a simple unequivocal expression in the realm of algebra even though models from ‘analytic geometry’ are used in the demonstrations. In other words, while algebra is useful as a tool in the demonstration of geometric statements it is not useful in formulating the statements themselves. The subject matter of the text lies here in what cannot be formulated in other terms or language, what cannot be otherwise said and what is not merely an example of something else. Hilbert’s subject matter in GdG is the systematic relationships that can be established between sets of axioms for geometry and the body of geometric propositions. These relationships, although demonstrated using algebraic analogies, are not simply reducible to algebraic statements and constitute a separate realm. Of particular interest to Hilbert is the extent to which geometry can be developed independently of the so-called axioms of continuity.28 These axioms introduce certain types of (logical) complexity into the

PROPOSITIONS AND PROOFS—THEOREMS AND PROBLEMS 37

geometries which model them. In particular, the second of these axioms (the axiom of completeness) is not a ‘first-order’ statement; i.e. it is not a statement about the individual elements that might make up a geometry but about all geometries which could be constructed with these elements and axioms. It is just these types of statements which led to the famous paradoxes in the then relatively new field of set theory and ended in the so-called ‘foundational crisis’. Hence the particular importance of Hilbert’s Chapter 6 in demonstrating that Pascal’s theorem is not independent of these axioms. While Hilbert demonstrates propositions about the theorems of geometry, the theorems of geometry are not the subject matter of GdG; they are not what the book is about. The propositions of GdG state relationships between axioms (or groups of axioms) and the geometries that they ground. Hilbert’s propositions are therefore about the consistency and independence of his axioms, not about geometry! His objective is to characterize relationships between axioms and geometries through the notions of consistency and independence. When GdG is put next to the Géométrie the pair of terms consistency/ independence can be seen to function in a manner parallel to the pair of terms geometrical/mechanical in establishing a subject matter for the text. Each of these pairs of terms serves both to distinguish the science of geometry (in the author’s conception) from other things and to structure the subject matter of this science. 2.8 The problem of geometric subject matter The approaches of Descartes and Hilbert to the problem of establishing subject matters for their texts can be summarized as follows: • Hilbert analyses logical relationships between propositions, Descartes finds methods for solving problems; • Hilbert characterizes types of geometries, Descartes constructs types of curves; • Hilbert demonstrates the existence of his subject matter but ignores the geometer; Descartes proves the existence of the geometer as thinker and cultivates his mind but does not demonstrate the existence of his subject matter. Yet in spite of these differences in approach Descartes and Hilbert share a common problem in establishing their subject matters. This problem can be stated in two ways, depending on whether one is looking from within the subject matter of a treatise or at the relations that these subject matters may have to other sciences. On the one hand, the pairs

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of distinctions geometrical/mechanical and consistency/ independence operate in both treatises within already given material. Neither of these approaches could operate without the assumption of previously known and given problems or propositions. On the other hand, although both authors deal with ‘geometry’, neither takes geometric propositions as his subject matter. Both organize and relate geometric objects, facts and propositions to other things so that the propositions which constitute them are (respectively) propositions about solving geometric problems and about the relations between geometric axioms and geometric propositions. Returning now to Euclid, the nature of his enterprise stands out in sharp contrast to those of these two later mathematicians for he is attempting to do what neither Descartes nor Hilbert did, i.e. to establish a subject matter made up of geometric propositions which does not and cannot rely on some previously given notion of what geometry is or might be. In stating that Euclid’s objective is the establishment of a subject matter for geometry I do not wish to suggest that geometric subject matters did not exist prior to the publication of the Elements. The evidence that does exist today, although scanty, suggests in fact that the standard title for geometric texts of the time was ‘Elements’. What can be seen of Euclid’s enterprise must be gleaned from within the text; our knowledge of his position vis-à-vis his predecessors permits only a hypothetical reconstruction of his relationships to them. Whatever models may have lain in front of him during its composition, it is clear that he intended his Elements to develop the science of geometry from the beginning, without reference to other mathematics. Put another way, Euclid was not attempting to discover new geometric methods or new foundations for geometry but rather a new science of geometry. To see what this could mean and how he proceeds in this connection the threads of the discussion from Chapter 1 must now be picked up. 2.9 Figures The series of definitions with which Euclid opens the Elements does not merely provide the individual elements or objects that will be scrutinized in his treatise. The sequence of definitions itself in its strict progress step-by-step uncovers an area to be explored in the ‘gap’ between lines-as-boundaries-of-surfaces and plane-surfaces-as-bounded-by-straight-lines in Definitions 6 and 7. Unlike Descartes on the one hand, who might propose a method for constructing boundaries (given some unit of measurement of lines) in response to the discovery of this problematic area, Euclid goes on to

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define boundary and figure in general in Definitions 13 and 14. But, unlike Hilbert, on the other hand, Euclid does not go on to set forth the general properties that characterize figures and boundaries, indeed in this generality they are not referred to again in the entire treatise. The peculiarities of Euclid’s procedure and, in particular, of these two definitions are highlighted by the above comparisons. To repeat earlier remarks, ‘figure’ in the Elements is a general context, a schema which can be particularized only by specifying its boundary, or, in other terms, by giving its definition. The boundary or definition is made up of parts which are merely the elements defined in Definitions 1–12 and which are now placed in new contexts. A modern mathematician may easily be puzzled by a definition of figure which does not provide material to be studied. Why not simply define circle, triangle and the rest directly? What does the general definition of figure and the statement that, for example, a triangle is a trilateral figure, add to our knowledge? The answer is certainly not to be found in what such a definition brings to any particular example of the general concept of figure. The knowledge of the different types of figure does not ‘add up’ somehow to a kind of inductive knowledge of figure in general.29 The value of the definition rather lies precisely in the provision of a context which can be specified through the twin operations of definition and delimitation. Even more forcefully, figure is the context in which definition and delimitation are the same thing, for to give a definition of a figure is to as give its boundary and to give its boundary is to give the figure ‘definition’ and as ‘boundary’). The notion of figure links the particular figures to be studied to the general problem of relating limitation to what is delimited in the case of surfaces. At the same time the notion of figure provides the method by which these examples of bounding are to be known, i.e. by specification of their boundary or, in other words, by definition. Note how this double functioning of figure/boundary differs from the double functioning of the pairs geometrical/mechanical and consistency/independence that were encountered in Descartes and Hilbert. The latter pairs operate between subject matters (already given or thought of as given) to cut out or to differentiate that which is to be examined. Euclid’s terms by contrast operate to establish a subject matter by linking a general problem to specific instances. These comparisons raise a further question: the Cartesian and Hilbertian distinctions geometrical/mechanical and consistency/ independence were located explicitly in the nature of the mathematical demonstrations used by their authors. What then does the Euclidean figure/boundary pair have to do with demonstration?

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2.10 QED/QEF: figures and diagrams It has already been noted that Euclid begins his series of propositions with a ‘problem’, i.e. in his formulation, a proposition that asks that something specific be done and closes with the conclusion that what has been requested has been accomplished (QEF). The ancient ruminations of the ‘problem of problems’ have also been noted. The first point to be grasped about Euclid’s use of the distinction problem/ theorem is that it operates within the genus ‘proposition’. While Descartes effectively conceived of constructions as being all of geometry and whereas Hilbert saw mathematics as a series of theorems (with problems acting only heuristically or as a second rate sort of definition), Euclid uses both problems and theorems to constitute his subject matter. This is simultaneously a reflection and consequence of the role of figures in his argument. On the one hand, a figure is a specific instance of boundary and being bounded so that by constructing figures through problems or QEF propositions Euclid is constructing ‘solutions’ to the general problem located by Definitions 6 and 7. On the other hand, a figure is also defined by its boundary so that by analysing their boundaries and the parts thereof Euclid can prove theorems or QED propositions about figures. The two types of propositions go hand in hand. Furthermore, as knowledge is accumulated, the nature of the relationship between the two types of problem shifts. The distinction of QEF and QED is to be found within the mass of mathematical propositions and is relative to it; it is not something which is imposed from without. But the notion of figure not only helps to grasp Euclid’s use of theorems and problems, it also helps to penetrate a far more notorious difficulty in understanding Euclid’s demonstrations, his use of diagrams. To see this, I recall Proposition 1 and its demonstration. This Proposition is the familiar construction of an equilateral triangle on a given straight line segment (see Figure 2.6). In commenting on the Postulates, the use of the passive perfect imperative verb form has been noted, as have the implications of its use. Here too in his construction in Proposition 1, Euclid uses this form of the verb, literally ‘let the circle BCD already have been described’.30 The construction is thus not about the drawing or making of the diagram (although this is precisely what Descartes’ discussion of geometrical/mechanical is about), it is about relating the parts of the already drawn diagram to one another. The distinction between Postulates and Propositions does not lie in the fact that one specifies or permits some constructional procedure (e. g. ‘ruler and compass constructions’) and the other employs it, rather it lies in the diagram itself. Euclid’s postulates are not propositions

PROPOSITIONS AND PROOFS—THEOREMS AND PROBLEMS 41

Figure 2.6 Euclid’s Proposition 1, Book I

which have been promoted to special status or selected as providing a simple and efficacious logical summary of the significant theorems of geometry as are Hilbert’s axioms. Rather they are specifications of the self-determination or self-sufficiency of the things defined in the definitions, the subject matter of the Elements. There are no diagrams associated with the postulates because the relationship between what is given and what is concluded is immediate. Even in the case of Postulate 5 with its complex formulation there is no mediation between the givens of the angles made by a transverse to two lines and the resulting point of intersection of the lines. By contrast, the relationship between the givens and the conclusion in propositions is not immediate. The relationship between the given line segment in Proposition 1 and the equilateral triangle constructed on it is mediated by the circles drawn from the endpoints of the line segment and the relationships between these circles and their parts (their radii). The diagrams are made possible by this mediation and their function is to exhibit it, i.e. to exhibit the relationships of figures and their parts. Either to overlook this functioning or to ask other things of the diagrams is to misunderstand the nature of Euclid’s demonstrations. Demonstrations in the Elements relate to the subject matter established in the Definitions as determined by the Postulates and Common Notions. The linkages between these ‘foundational statements’ and the demonstrations are provided not by general logical equivalences as in Hilbert but in the specific relationships established between the parts of the things which have been defined. Euclid does not carefully define his material only then to render the specific characteristics of the definitions irrelevant. The process at work in the

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definitions, the focus on parts and wholes, boundaries and boundeds, is also at work in the demonstrations. Hence although one may raise questions of, for example, a topological nature about Euclid’s diagrams,31 such questions cannot be formulated in the language of the Elements. This should not be considered either a strength or a weakness; such questions can indeed be addressed in other sciences, but not in the science of geometry as set forth in Euclid’s text. In particular, Euclid has no way even to pose the question ‘do the circles (in the diagram for Proposition 1) intersect?’, even though mathematicians from ancient times have reproached him for not doing so. The objectors claim that his science is faulty in employing demonstrations which are not entirely ‘rigorous’ and that in more complex cases it is relatively easy to construct diagrams which can lead to false conclusions through so-called ‘geometrical sophisms’. On this view of things (very closely related to Hilbert’s) the science itself must operate so as automatically to exclude the possibility of error and thus arrive at certainty.32 The contrasting Cartesian view is to cultivate the mind of the geometer, locating certainty in the method practised by such a mind and not in the knowledge accumulated. Euclid takes neither position since both involve a notion of certainty which is alien to his approach. His science has a subject matter and a method which follows from the nature of the subject matter; to ask for certainty beyond this would be chimerical at best. Matters of first philosophy are clearly involved in this discussion which go well beyond the framework of a commentary on the Elements. That these philosophical issues are not more widely discussed reflects the rather high level of dogmatic acceptance of certain philosophical tenets within the mathematical and scientific community. Given an enterprise of the sort that Hilbert undertakes (recall his words ‘the logical analysis of our concept of space’) it is indeed appropriate to incorporate axioms of continuity or similar statements to deal with this lacuna as well as others that are alleged to arise in Euclid’s demonstrations. Of course to do this the axiomatizer must take the science to be axiomatized as already given. Euclid on the other hand is going about the establishment of a subject matter in the first place. Just because of this he would not be able to proceed along the lines Hilbert proposes. To repeat, when examining the ‘rigour’ of Euclid’s demonstrations the nature of his enterprise must constantly be borne in mind. This is not because he is incapable of following the thought processes of Hilbert, indeed his text the Data may be read as providing a sort of ancient equivalent of Hilbert’s style of axiomatization. The Elements, however, focuses on the problem of establishing a geometric subject matter in which these questions cannot and need not be posed.

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To bring home the character of Euclid’s demonstrations it will be useful to refer to Proposition 4, the first of the ‘QED propositions’ in Book I and one which has provided for most readers, on the evidence of the accumulated commentary, an excellent example of a mathematical demonstration gone awry. The proposition states the ‘side-angle-side’ criterion of congruence of triangles from elementary geometry relying on the method of ‘superposition’. The first point to be noted here is that for Euclid superposition is not a method of construction. This is simply because it is not referred to in the Postulates. We shall see that Euclid’s use of superposition lies in comparing two figures not in constructing one ‘on top of’ the other. Second, the specific formulation of the role of superposition as set out in Common Notion 4 should perhaps be read ungrammatically as stating that ‘Things which are superimposing are equalling to each other’. The conclusion is that the things are equal, or in the case of Proposition 4 that the triangles themselves, not their parts or their ‘areas’, are equal. The other ‘congruence’ theorems in Book I (Propositions 8 and 26) are not about triangles but about their parts and hence, as many commentators have noticed, their demonstrations do not rely on superposition. Although some have supposed that this is due to some wariness on Euclid’s part in using superposition,33 we may also hypothesize that Euclid does not view superposition as the right tool for the other cases where the objective is not to compare the things themselves but rather their parts. In Proposition 4 on the other hand, Euclid’s method is to compare the triangles by comparing their parts (see Figure 2.7). First the point A is compared with point D. There being no parts to points they merely coincide. Line AB is then compared with line DE. The conclusion is that the endpoints coincide. Why? Because as previously stated, points are the extremities of lines, i.e. points are that by which the length of a line is indicated. Now it is assumed by hypothesis that AB and DE have the same length. Were they not to have the same endpoints then there would have to be some other determinant of the length of a straight line besides its extremities. But the whole point of the Postulates in general and Postulate 1 in particular is to make straight lines self-determining things, hence the endpoints of the two line segments must coincide. This pattern of argument can be applied to the remainder of the demonstration. For example, the sides BC and EF must coincide or something other than the lines forming an angle would be involved in the determination of the angle. Now the most troublesome part of this demonstration has always been the conclusion that the remaining sides AC and DF coincide as well. But this follows in the same way. It having already been determined that the endpoints of these lines coincide, were there to be

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Figure 2.7 Euclid’s Proposition 4, Book I

the possibility of a different line segment between the endpoints this would mean that Postulate 1 had not succeeded in making lines self-determining. If two such lines were possible, in applying Postulate 1 how would we know which of the lines required by the Postulate had actually been drawn? Some form of external determination would be required, which, as has been stated repeatedly, is exactly what the Postulates are designed to avoid. It should go without saying that there are almost endless possibilities for mathematical sciences which choose to supply one or more of the types of ‘extra determinants’ that Euclid rigorously excludes. These alternative approaches can and do use axiomatic, group theoretic, differential geometric or function theoretic (or indeed other) methods to provide determinations for notions such as length, area, position or angle in a manner which is alien to the Elements. Euclid has not chosen arbitrarily to work within the confines of a given ‘geometry’ or type of geometry. Rather he has chosen to undertake a particular enterprise, the establishment of a geometric subject matter. It is this decision which dictates the course of his argument. 2.11 Overview of Book I A brief overview of Euclid’s procedure in the course of Book I will show how the argument develops from this point forward.

PROPOSITIONS AND PROOFS—THEOREMS AND PROBLEMS 45

The subject of Book I is triangles and Euclid begins, as noted above, with the construction of an equilateral triangle as Proposition 1. Note that at this point Euclid’s argument does not provide him with the means either to: • relate the line segment on which the triangle is constructed to the triangle itself; or to • relate triangles constructed on different line segments. Similarly, Proposition 4 provides a criterion for determining whether two triangles are equal. At this stage in the argument Euclid cannot: • relate the parts of the triangles to the triangles; nor • construct triangles in such a way as to be able to determine whether or not they satisfy this criterion. Book I systematically develops the means for relating triangles to each other and for relating the parts of triangles to the triangles themselves. Table 2.2 outlines the entire sequence of propositions in Book I and groups them according to function. A few general remarks on the structure follow. Euclid alternates theorems, stating contexts or circumstances in which the parts of triangles or other figures determine each other or in which the parts of two triangles or other figures can be related, with problems which take advantage of these determinations to construct something. Proposition 22 is a milestone in the process showing the construction of a triangle out of three given line segments which satisfy only the relationships among themselves that the sides of a triangle must have (cf. Proposition 20). This construction in turn provides a complete determination of the relationships of equality and inequality that must hold between the parts of two given triangles as expressed in Propositions 24–26. The introduction of parallel lines in Proposition 27 has as its objective the study of parallelogrammatic areas in Proposition 34 and following. Triangles are rigid (Proposition 7) and therefore show rich internal structure and satisfy useful criteria for equality of themselves and their parts. For the very same reason they are hard to work with. In Proposition 35 Euclid begins to show how parallelogrammatic areas can be related to one another independently of their specific ‘shape’. The payoff of this analysis is Propositions 37–39 relating triangles to one another without use of the ‘congruence conditions’. The next group of Propositions (41–44) provides the ability to construct parallelogrammatic areas equal to given figures. The last ‘problem’ (Proposition 45) provides for the construction of a square on a given line segment (compare with Proposition 1).

46 FIGURES OF THOUGHT

Table 2.2

The finale of Book I is the Pythagorean theorem and its converse. With these statements the objective is achieved of demonstrating a determinate relationship between a triangle and its parts.34 Within the context of a specific type of triangle (right triangles) parts of the boundary of the triangle can be placed in a very specific relationship to one another simply by virtue of being the parts of the boundary of the triangle. The specific relationship demonstrated is of interest because it relates lines to figures they bound and to squares and suggests a broader context (to be called ‘magnitude’ in Book V) into which they

PROPOSITIONS AND PROOFS—THEOREMS AND PROBLEMS 47

might be placed. The Proposition also serves as a stimulus for the investigations in Book II where similar types of relationships are established between the parts of other types of triangles. Finally an examination of the relationships between the sides of right triangles leads to the problem of incommensurability which is the subject of Book X.35 Euclid thus directs the Propositions of Book I to a vantage point from which a great deal of the remainder of his treatise can be viewed. The sequence of propositions should not be read backwards from the objective however. In constructing his subject matter Euclid is led at all times by what has been demonstrated up to a given point and the questions that arise therefrom. The interaction of problem and theorem allows him to progress his argument from point to point, accumulating both knowledge about the figures and capabilities to construct them. This sequence of propositions should be read as the ‘plot’ of the book. 2.12 Summary Euclid’s figures occupy a place between construction and demonstration, QEF and QED. Both Descartes’ methodical constructor and Hilbert’s formalized science in their single-minded determination to press a particular approach may appear to be more rigorous than Euclid’s alternation of problem and theorem and his use of diagrams. But Euclid’s subject matter is indeed the propositions of geometry and he constitutes it from the beginning without reference to previous subject matters or other mathematical facts. Descartes and Hilbert in one sense already know the terrain they wish to cover and are simply seeking efficient means to cover it. Euclid’s method is adapted to the constitution of a new scientific subject matter, an enterprise which is perhaps less spectacular than the discovery of new scientific data, facts, methods or principles but one which underlies all of them. Further general comments on the philosophical import of Euclid’s approach are postponed until the final chapter.

48

Part II The development of methods of measurement in Euclid, Dedekind and Kronecker

50

3 The contexts of measurement

3.1 Introduction In Part I the subject matter of the Elements has been shown to consist of figures which are the most general type of ‘bounded thing’. Specific figures then provided instances of the essentially indeterminate relationship discovered between limited plane surfaces and their boundaries by the Definitions of Book I. This subject matter is expressed in propositions which take the form of both ‘constructions’ and ‘demonstrations’. By contrast the subject matter of Descartes’ Géométrie was seen to be a certain kind of line or curve (so-called ‘geometric’ curves as distinguished from ‘mechanical’ ones) expressed in propositions which are principally constructions, while in Hilbert’s Grundlagen der Geometrie the subject matter is the logical relationships between axioms and theorems as expressed in propositions which are principally demonstrations. Euclid establishes his subject matter in three steps: • he sets out a series of definitions which proceed rigorously, step-by-step, through the aspects of the subject matter; • he requires, through a series of postulates, a type of integrity or self-determination for the elements of the subject matter; and • he provides possibilities for relating the elements to one another in a series of common notions. The Definitions, Postulates and Common Notions of the Elements neither describe the subject matter of the book nor do they construct it out of primitive undefined building blocks. Rather these opening or ‘foundational’ statements provide a way to ‘read’ or interpret diagrams which relate the parts of figures to one another. Propositions in the Elements relate the parts of figures in constructions and

52 FIGURES OF THOUGHT

demonstrations, and the interplay between these two modes of statement locates the specific aspects of Euclid’s subject matter. Neither Descartes nor Hilbert followed Euclid’s lead, as neither of these later mathematicians was concerned to found1 a mathematical science but rather sought to treat an old science in new ways. Descartes views geometry as a series of given problems to be solved and as an example of the method that the mind can use to solve any problem in any science, while Hilbert’s objective is to provide a foundation for geometry, i.e. a set of logical relations between axioms and propositions which demonstrate how axioms are consistent with each other and how propositions depend on sets of axioms. For both Descartes and Hilbert there is a subject matter of geometry which ‘pre-dates’ their effort and which provides a kind of substrate to it. This common assumption (albeit of very different ‘geometries’ as substrata) allows Descartes to show how specific constructions can be performed and Hilbert to demonstrate how particular logical relationships obtain. Extracting the essential aspect of Euclid’s approach when it is compared with Descartes’ and Hilbert’s, it may be said that Euclid provides a context in which figures can be interpreted. This enterprise is to be contrasted with that of Descartes, on the one hand, who sets out methods for the construction of curves (or lines) and that of Hilbert, on the other, who sets out methods for the demonstration of logical relationships. The context in which Euclid interprets figures in Books I–IV of the Elements is set out in the very first definition of Book I, the context of ‘part’. ‘Part’ may be said to provide a context for Books I–IV because, while it is used to define the elements of the subject matter, it is itself not defined nor determined. The word takes on a variety of meanings in the Definitions in Book I, making a single, unambiguous definition impossible.2 The use of a common ambiguous term in the various definitions provides a means of relating them to one another, in other words, a context. In the first definition of Book V Euclid provides a definition for ‘part’ (five books after its introduction!) signalling a change in the context within which he is operating. To appreciate the significance of this alteration of context it is necessary to review briefly the position of Euclid’s argument at the end of Book IV. The final proposition of Book IV provides for the construction of a regular figure of fifteen sides by means of its inscription within a given circle. There is an echo in this proposition of the first proposition of Book I demonstrating the construction of an equilateral triangle on a given line segment. This similarity between the two propositions allows for a comparison (sets them within a context) which forcefully brings out the differences between them. Not only is the later construction much more complex (involving many more parts) but it also takes place

THE CONTEXTS OF MEASUREMENT 53

in a more determinate setting. The equilateral triangle of Book I is constructed almost in a vacuum. Given that this construction is the first proposition of the Elements, Euclid has perforce not previously developed any way to relate the triangle to other figures or to relate two such triangles to each other. The great ‘success’ of Proposition 1 of Book I is merely the construction of the triangle. By the time Euclid arrives at Proposition 16 of Book IV he has developed a range of ways in which figures can be related to one another. The entirety of Book IV is devoted to constructions which show how two figures can be related to each other by inscription and circumscription. Since Euclid’s geometry is not a geometry of ‘space’ in which sets of ‘co-ordinates’ can be used to relate figures to one another, inscription and circumscription are the only ways in which figures can be related to one another as figures of a given type.3 The circumscribed and inscribed figures are related not only as figures but through their parts as well. The radii of the inscribing and circumscribing circles relate to the sides of the figures so that these circles provide, in a very concrete sense, contexts for the figures. The elaboration of these contexts measures the progress that Euclid has made from Proposition 1 of Book I to Proposition 16 of Book IV. This progress also demonstrates its limitations. Although the regular polygons can be compared with the circles to which they are related, Euclid has no way to relate to each other the polygons constructed in relation to different circles. Put another way, he can compare two figures in a pair, one with the other, but he has no way of comparing two such pairs of figures. Book IV represents the limit of Euclid’s ability to relate figures to each other using the tools provided by the context of ‘part’ as developed in the Definitions, Postulates and Common Notions in Book I. Further progress requires a new context. 3.2 Part and measure At the outset of Book V4 Euclid signals a turning point in his argument by returning to the terminology of Definition 1 of Book I. It will be recalled that this definition stated that, A point is that which has no part. Parts provide the context in which other things can be discussed and this first definition establishes a point as that about which nothing can be said, other than that it is. When this definition is compared with the other definitions we see that all other things do have parts. Lines for example have one part, length; surfaces have two parts, length and breadth; while other things such as angles or figures are defined in terms

54 FIGURES OF THOUGHT

of their parts. ‘Part’ provides the context for all these definitions and therefore cannot itself be defined. Now, some five books later, Euclid provides a definition of ‘part’. His definition uses other undefined terms, ‘magnitude’ and ‘measure’, signalling a new context for his discussion, and runs as follows: A magnitude is a part of a magnitude, the lesser of the greater, when it measures the greater. This Definition should be taken with the one which follows, The greater is a multiple of the less when it is measured by the less. Two terms are thus defined, part and multiple, by means of the relationship of ‘measuring’ between ‘magnitudes’. The difference in the definitions lies in the active and passive voices of the verb ‘measures’. This procedure has proven quite baffling to most commentators, who have mostly chosen to pass over these two definitions in silence.5 The new context of Book V is ‘magnitude’. As with ‘part’ in Book I, magnitude is not defined in Book V. This is because magnitudes in the Elements are not individual things or classes of things which are subject to specific definition but rather the result of viewing already defined things in new ways. Recall that in Book I Euclid equated ‘definition’ with ‘boundary’ through the Greek word ‘horos’ which can be used with either meaning. To define a magnitude in Euclid’s sense (were this to be possible) would mean to provide some kind of constituting limitation as, for example, the tripartite boundary that defines and constitutes a triangle. This cannot be done in general. In the wake of the first two Definitions in Book V, where already defined things are related to each other as ‘part’ and ‘multiple’, i.e. as measure and measured, we are dealing with these things as ‘magnitudes’. This is a new context in which to view already defined things but it does not, in and of itself, provide new material or new elements. The measuring is relative, each thing measuring other things, and is not subordinate to a single standard of measurement any more than the figures of Books I–IV were located or placed in any single space.6 The verb ‘to measure’ here is, in a sense, a generalization of the verb ‘to equal’ as previously employed in Books I–IV, so that a review of this earlier notion will help to clarify the newly introduced concept. Throughout the Definitions of Book I, in defining straight lines, plane surfaces, right angles, circles and various types of polygons, Euclid makes use of a notion of equality. As noted previously, his approach is likely to appear curious to modern eyes in that he does not provide, for example, a general notion of measurement for angles and then declares that two angles are equal if they have the same measurement. For Euclid the measure and thing measured are inseparable and are defined together. Again, for example, by Definition 10 of Book I, right angles are

THE CONTEXTS OF MEASUREMENT 55

angles which, in certain geometric circumstances, are equal. Postulate 4 then requires that all such angles be equal to each other. This provides right angles with a degree of self-determination since no specification apart from that provided in the definition is required to determine them completely. Equality (of angles in this instance) is the basis of all measurement (of, for example, angles) and is prior to other measures. Although itself not susceptible of definition, it can be used to define other relationships. Similarly, in Book V, the verb ‘to measure’ is not defined, nor is there any attempt to show how to demonstrate that in fact one thing measures another. The relationship of ‘measures’ is constitutive, one magnitude measures another if the second is ‘made up’ of the first as its parts, just as two equal angles constitute right angles. No specification of how to go about the measuring is either possible or necessary and the relationship of measuring is possible whenever and wherever the relationship of equality is. In this sense we could say that measure generalizes equality. The difference is that the multiple can be made up of or measured by the part any number of times. By contrast, in definitions such as that of right angle or equilateral triangle a specific number of parts were involved. As a relationship equality is ‘bi-directional’ or ‘reciprocal’ in nature, if a equals b then b equals a. When equality is generalized to measure, this reciprocity is enhanced. It is possible to speak of measure and measured and therefore to use the relationship in ‘two directions’. Whereas the relationship of equality is the same in ‘either direction’, the relationship of measuring shows two sides, active and passive, measures and measured, part and multiple. The full context of magnitude is located by the verb forms measure and measured or, in simpler noun form, part and multiple. The previous uses of equality were special cases of the relationship of measuring just as the examples of limited lines, surfaces and angles were special cases of the general notion of ‘boundary’ in Book I. The construction of regular polygons in Book IV indicated the limits of the context of ‘part’ in dealing with a large number of parts simultaneously. Definitions 1 and 2 of Book V move from the large but determinate number parts dealt with in Books I–IV to an indeterminate number parts as controlled by the relationship of measuring. The purpose of the generalization from part to measure is not to treat of ‘magnitudes as such’ or ‘in general’, just as Euclid never treats of ‘figures as such’. The new context rather provides new ways in which previously defined things can be viewed. By defining ‘part’, this notion itself can become the object of study rather than merely providing the terms through which other things can be discussed.

56 FIGURES OF THOUGHT

It should be noted that at this point in Euclid’s argument, the nature of the relationship ‘part and measure’ is left open in two respects: • Euclid has no way to specify ‘how many times’ one magnitude measures another; and • Euclid has no way of determining whether one magnitude in fact measures another. The solutions to these problems go beyond the mere introduction of the new context of magnitude and are addressed in Books VII–IX and in Book X respectively. Returning to Book V, the introduction of the new context of magnitude itself raises two interrelated problems. First, as noted above, there is no way to supply a constituting limitation to ‘magnitude’ since it is not made up of anything in particular. Lines are constituted by length, i.e. by one part, so they are bounded by points which have no parts. Similarly surfaces are limited by lines. But magnitudes are not anything in particular, they are the things previously defined now regarded in a new way. Since to define is to bound or delimit in the Elements it is entirely unclear how Euclid will be able to make definitions respecting magnitudes. Second, since magnitudes are not anything in particular, what is the relationship of demonstration and construction to be? How can magnitudes be constructed? Put another way, there are no Postulates associated with Book V, i.e. there are no requirements set forth which provide the Definitions of Book V with the kind of self-determination that was seen in Book I. Such determination as they require will have to come from that previously supplied. How then does Euclid go about demonstrating propositions in this new context? 3.3 Ratio and proportion Definitions 3–6 of Book V, the definitions of ‘ratio’, ‘to have a ratio’ and ‘to be in the same ratio’ or ‘to be proportional’, provide structure to the notion of magnitude by providing sets of relationships which can be studied and manipulated independently without recourse to a prespecified notion of measurement. Definition 3, of ‘ratio’, states that a ratio is a ‘type of’ relationship between magnitudes of the same kind in respect of size. The function of this definition is to place the magnitudes of Definitions 1 and 2 into relationship with each other. Things become magnitudes when they are viewed in the context of measuring or being measured by each other. Two magnitudes are of the same kind (as magnitudes) when the context

THE CONTEXTS OF MEASUREMENT 57

in which they are being viewed as magnitudes is the same, i.e. when the things they measure and are measured by are the same. A ratio is any relationship in respect of this reciprocal measuring and being measured on the part of two magnitudes. By being put in a ratio the magnitudes ‘leave behind’, as it were the parts, and multiples that allowed them to be considered as magnitudes in the first place and are related solely and directly to each other.7 Definition 4, ‘to have a ratio’, specifies a particular way in which a ratio can be demonstrated, namely that each of the magnitudes have multiples which exceed each other, i.e. can be compared with each other. In Books I–IV it is the things themselves which are compared with each other as being equal to, greater than or less than. In Book V it is not the things themselves which are compared, but other things to which the originals bear the relationship of part or multiple. This is the true sense in which magnitude is seen as a generalization of the previous material and clearly provides an enormous expansion in the power and extent of the comparisons that can be made. According to Definitions 1 and 2 of Book V, it could be said that magnitudes do not come ‘by themselves’ or singly but rather come as things which are surrounded by the hosts of other things that they measure and are measured by. Their nature as magnitudes is wholly relative to this measuring activity. In the context of ‘ratio’ and ‘having a ratio’, the magnitudes leave behind the particular measurings which allowed them to be considered as magnitudes in the first place and relate directly and solely to another magnitude through the ratio. Euclid has no reference here to notions such as ‘continuity’ or ‘the axiom of Archimedes’.8 These ideas all presuppose a fixed notion of magnitudes (such as ‘the real numbers’) which are, contrary to Euclid’s approach, particular entities defined as magnitudes. The consequences of thus entitizing the notion of magnitude will be explored further when the mathematical ideas of Dedekind and Kronecker are compared with those of Euclid. Definition 5 completes this sequence by specifying when two ratios are the same via the famous test of equimultiples. This Definition has been much commented upon due to its complexity and does not require a long commentary here in view of what has gone before. It should be noted, however, that Euclid explicitly does not speak here about equal ratios but about magnitudes being in the same ratio.9 Equality is a relationship among specific measured things; ratios are not measured things or magnitudes but rather relationships between magnitudes.10 Nor should it be thought that Definition 7, which uses the equimultiples test to define the phrase ‘to have a greater ratio’, licenses a view of ratio as a kind of magnitude in itself. In the earlier books it has been noted that Euclid nowhere defines the relationships of ‘equal to’ or ‘greater or

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less than’ in the cases of specific measured things such as lines or angles. In the Common Notions he does provide ways of connecting already measured things in relationships of equality (through superposition) and inequality (through part/whole) but these only provide circumstances under which things can be compared, not definitions of what it means for a line or an angle to be equal to or greater than another. Such definitions are not possible in Euclid’s system. Paradoxically, the very fact that Euclid defines the phrases ‘in the same ratio’ and ‘in a greater ratio’ here implies that ‘ratio’ is not a measure or a magnitude! Euclid brings out the similarity between ‘in the same ratio’ or ‘proportion’ and ‘figure’ as structured contexts in which previously given material takes on new functions in Definitions 6 and 8 by calling the magnitudes which are in proportion the ‘terms’ (‘horoi’, limits, boundaries or definitions) of the proportion and by stating that a proportion must have at least three terms.11 The ‘terms’ of a proportion limit or define a proportion in analogy to the boundary of a triangle which limits and defines a figure. The subsequent definitions operate by treating magnitudes as they appear in proportions just as Definitions 15–22 of Book I define figures in terms of their boundaries. The structured context of proportions provides a means to make definitions respecting magnitudes. This is the answer to the first of the two questions posed above; the second question, pertaining to the mode of demonstration, will now be addressed. 3.4 Books V and VI—propositions about proportions As noted previously there are no Postulates (or Common Notions) associated with Book V and the theory of magnitudes in proportions. The reason for this once again is that magnitudes are ‘not anything in particular’ but rather a new way to regard already known things. The determinations and constructions which have been established previously for particular figures and their parts will continue to serve in those specific cases, but there is no new material on which to impose additional requirements or specifications at this point. Book V contains no constructions (in direct contrast to Book IV) but is made up entirely of theorems. A further consequence of the nature of magnitudes is that the use of diagrams shifts. Throughout Book V (and beyond), Euclid uses line segments to illustrate his magnitudes but at no time do the properties specific to line segments (their ‘geometry’ as it were) enter into the demonstrations. A brief look at Proposition 1 of Book V will illustrate the procedure. Here Euclid is demonstrating that if two series of

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magnitudes (indefinite numbers of magnitudes) are such that the magnitudes in the first series are all the same (unspecified) multiple of the corresponding magnitudes in the second series, then all of the magnitudes in the first series added together are the same multiple of all of the magnitudes in the second series added together. The diagram illustrates this proposition with a case involving only two multiples in each series and with the multiples in the first series being doubles of the multiples in the second, but the particular case illustrated is treated as entirely general. The key step refers to Common Notion 2 from Book I, equals added to equals are equal. The Common Notion is of course not limited to line segments but is ‘common’ to all measured things. It is not line segments that are being added but magnitudes that are being compared; the diagram just simplifies the presentation. This is completely different from the use of diagrams in Books I–IV as discussed above in Part I. Further differences from the earlier books can be seen, for example, in Proposition 11 of Book V, where Euclid proves the equivalent of Common Notion 1 for ratios; two ratios which are the same as a third ratio are the same. This shows again that ratios are not measured things or magnitudes since the Common Notion does not apply to them as such, only to their terms.12 Turning to the sequence of Propositions in Book V, the first six Propositions deal with the relationship ‘a is the same multiple of a
as b is of b
’, the so-called relationship of ‘equimultiples’. Given the importance of this relationship to the Definition of ‘in the same ratio’, these Propositions should be seen as forming the building blocks for the argument of Book V. Table 3.1 provides an analytical layout of this group of Propositions. The substitution of algebraic notation for the statement of these Propositions, such as provided by Heath in his notes, although it makes some of the sense more accessible to the modern reader, is dangerous because it dissolves the relationship of equimultiples into a series of equalities. Modern mathematics treats each of the magnitudes separately whereas Euclid’s argument sets them in relationship to each other in ratios and then sets the ratios in relationship in proportions. If we focus then on the relationship of equimultiple the logic of Euclid’s formulations becomes clearer. For example, Propositions 1, 2 and 5, 6 are seen to form a parallel set of pairs, dealing respectively with addition and subtraction and the relationship of equimultiple. This type of grouping of propositions in parallel pairs is seen again (with some modification) in Book VII. In Propositions 7, 8 and 9, 10 (Table 3.2) Euclid provides another pair of pairs of propositions, here relationships between magnitudes are shown to lead to relationships between ratios and conversely.

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Table 3.l

Note that in this group of propositions we encounter phrases such as ‘Equal magnitudes have to the same the same ratio’ and ‘Magnitudes which have the same ratio to the same are equal’. Each of these propositions is formulated in terms of a given magnitude against which other magnitudes are compared using a phrase with the structure ‘ [have a given relationship] to the same’. The next group of Propositions, 11–15, also relates magnitudes in ratios to the ratios, going back and forth between specification of the magnitudes leading to relationships between ratios and then the inverse process (Table 3.3). Here, however, the relationships follow from the ratios themselves and comparison with a given magnitude ‘outside of the ratio’ is not necessary. These propositions have internalized the relationships between the magnitudes which ‘make up’ a proportion and the proportion they make up. From this point onward Euclid does not need to refer to magnitudes outside of the ratios so that the ratios can be studied in themselves alone.

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Table 3.2

Propositions 16–19 then study various ways in which ratios (as opposed to the magnitudes which make them up) can be manipulated. In these Propositions the structure of the relationships between the magnitudes is altered but the ratio stays the same (Table 3.4). Propositions 20–25 (Table 3.5) complete the arguments from magnitudes to ratios and back aga in, now including the relationships of greater than/less than as well as equality and using further techniques of manipulation of proportions culminating in Proposition 25 which gives a specific relationship between magnitudes in a proportion not relying on external hypotheses and relating simply to themselves, not to an outside magnitude. Book V provides a structure of argument which is parallel to that presented in Book I in that the existence of a relationship between parts and that which they are part of (parts of a figure in Book I, terms of a proportion in Book V) is used to establish relationships between the parts themselves. The crucial difference between the two lies in the fact that there are no constructions in Book V because magnitudes are not entities in themselves but rather new ways of looking at other things. The argument of Book V has not been without its critics, ancient and modern. The classic criticism of the book centres around Proposition 18 and the existence of the ‘fourth proportional’,13 i.e. if magnitudes a, b, and c are given, can a magnitude d be found such that a:b::c:d? It is alleged that Euclid merely assumes the existence of such a fourth proportional in his proof of Proposition 18 without any justification and that an axiom of some kind is required to validate this assumption. This criticism arises from a failure to take full account of the importance of not viewing magnitudes as ‘particular things’ but rather as already known things viewed in new ways. There are two (related) difficulties with this criticism. In the first place, to attempt to provide a general axiom or postulate which would guarantee the existence of this fourth proportional would be to make specific entities (of one kind or another) of magnitudes and thus to go entirely against Euclid’s approach in Book V. Second, and as a natural consequence, Euclid has no way to construct magnitudes ‘in general’ or to determine their existence. Naturally this may be possible in other mathematical systems

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Table 3.4

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(the consequences of including the ability to make such determinations will be explored later), but not in the Elements. Modern readers have also been puzzled by the logical structure of Book V.I.Mueller,14 summarizing the views of scholars who have studied the deductive structure of the book, comments that as a deductive sequence it poses a number of puzzles inasmuch as many of the propositions are not used either in later sections of the book or in other parts of the Elements. In part the puzzlement arises from the attempt to read Book V as a ‘general’ treatment of magnitude and ratio and as a theory of ‘irrationals’ which then is applied in specific cases in Books VI (plane figures) and Books VII–IX (numbers or discrete magnitudes). When the point of view is adopted that Book V is a study of certain kinds of relationships between already known things which permit a wider range of comparisons to be effected, its role in the Elements becomes somewhat clearer. This same point of view also clarifies the internal structure of the book. As presented here Book V is a systematic exploration of the various ways in which relationships can be established between magnitudes in ratios and the ratios themselves. The motion of the argument is in both directions—magnitudes to ratios and ratios to magnitudes – a type of argument which is not repeated in the books dealing with ‘number’ for reasons which will appear. The process commences with the relationship of equimultiples which underpins ratio and proportion and proceeds from external to internal relationships. The culmination of Book V is a categorical statement of a relationship which holds between magnitudes in a ratio by virtue of the existence of the ratio itself, much as Book I ended with the Pythagorean theorem. Whether this argumentative sequence is visible in the deductive structure or not, it is surely a plausible explanation for the structure of the book as we find it.15 Book VI applies the theory of ratios and proportions to the plane figures of Books I–IV and develops constructions and theorems which are related to the earlier books but now have added powers of discrimination and determination. In particular, Euclid is able to compare figures and their parts and to develop ratios between these (note particularly Propositions 20 and 23). We shall leave Book VI aside at this point, however, to go on to Books VII–IX and Euclid’s approach to ‘number’. 3.5 Magnitudes and multitudes Having developed a theory of ratio and proportion based on magnitude in Books V and VI, Euclid apparently returns to the same subject in

64 FIGURES OF THOUGHT

Book VII and builds a new theory based on ‘number’. The reasons for this double treatment of the subject and the relationships between the two treatments have remained obscure although the applications to the subject of ‘incommensurables’ in Book X are evident.16 The principal reason for the obscurity surrounding Euclid’s procedure is the failure to appreciate the full import of the point which has been repeatedly stressed in dealing with Book V, that Euclid does not view magnitudes as particular entities in themselves but as new contexts in which to view previously known and defined things. Magnitude was structured as a context by placing measured things in relation to one another. In Book V these relationships were determined through the notion of equimultiple, one magnitude being the same multiple of a second that a third is of a fourth. No method was necessary or available for specifying the relationship of multiple beyond ‘the same as’. By contrast, Book VII provides a means of studying questions of the form ‘how many times’ one magnitude is a multiple of another and thus forms a necessary complement to Book V.17 For this purpose the context of Book V is altered. Euclid is now no longer concerned with relating magnitudes to each other as magnitudes, hence in Definition 1 of Book VII he fixes or ‘freezes’ a constituting magnitude by abstracting all qualities from it except that of its existence, this is the concept of ‘unit’. A unit is that by which each existing thing is called one. Anything is a unit when it is looked upon as one single thing. This definition provides another echo of Definition 1 of Book I which specifies that anything can be seen to be a point provided it is viewed as having no parts. While Definition 1 of Book V relates back to the term ‘part’ in Definition 1 of Book I and makes it into the basis for the concept of magnitude, Definition 1 of Book VII focuses on the ‘is’ of Definition 1 of Book I and makes it the basis for the notion of multitude or number. A key point is the use of the phrase ‘is called’. The definition of unit is verbal and no more constitutes some specific thing than does the concept of magnitude; as such both differ from the definitions of points, lines and angles. A unit is that which results from a particular way of regarding already defined things, namely regarding each of them as an existing thing. Definition 2 then runs: A number is a multitude composed (‘put together’) of units. Numbers are just collections of units. In Euclid’s approach there is no sequence of numbers nor are numbers one specific type of thing, rather units are things viewed as individuals and numbers are collections of units. While magnitudes are already defined things viewed through the essentially relative relationship of part and multiple, numbers are assembled by regarding existing things as units and do not require

THE CONTEXTS OF MEASUREMENT 65

relating to other numbers to achieve definition. Thus ‘multitude’ is defined in Book VII whereas its apparent counterpart ‘magnitude’ is not defined in Book V. Such parallels as exist between the sequences of definitions in Books V and VII are located by tying the notion of ‘part of’ in Book V, which is a relationship between two magnitudes to that of ‘unit’ in Book VII, which is a relationship between a thing and itself. Having stressed the distinctions between magnitude and number, it must be noted that numbers can be viewed as magnitudes as well. To show this Euclid uses the relationship of ‘measures’ which constitutes magnitudes and hence (further complicating his extraordinary use of the term!) he goes on to define a notion of ‘part’ for numbers (Definition 3): A number is a part of a number, the less of the greater, when it measures the greater. Of course, this is the definition of ‘part’ as given in Definition 1 of Book V word for word. The literal identity of the definitions only highlights the differences in their meaning and function. Magnitudes are constituted by the relationships of measuring and being measured whereas numbers are made up of units. Nonetheless, in so far as a number is considered to be a magnitude, the relationship of ‘part’ is the same for both. This is of great importance in Book X. In speaking of ‘lesser’ and ‘greater’ here Euclid is not referring to concepts of ordinality or cardinality which have been used to characterize ‘numbers’ in mathematics since Cantor and Dedekind. As noted previously, Euclid nowhere defines equals, greater than or less than for magnitudes or measured things. Simply put, without these concepts as givens there is no concept of magnitude or measured thing for Euclid. If numbers are treated as magnitudes then they of course will be susceptible of these relationships and, in turn, these relationships can be no more defined for number in terms of other things than would be the case for lines or angles. The significance of Definition 3 is perhaps best seen when it is taken together with Definition 4, but parts’ when it does not measure it. There is no definition of parts in Book V and no consideration of the situation where one magnitude does not measure another. The reason is that the reciprocal relationship of ‘measures-is-measured-by’ which constitutes magnitudes has no parallel in the case of numbers. Numbers are ‘put together from’ units in a ‘one directional’ way whereas magnitudes are entirely relative to each other, the relationship of ‘measures’ being balanced by the relationship of ‘measured by’. The active and passive formulation of Definitions 1 and 2 of Book V are thus related to the singular and plural formulations of Book VII. Definition 4 of Book VII has no parallel in Book V because magnitudes come about solely as a result of the relationship of measuring and being measured;

66 FIGURES OF THOUGHT

there is nothing to say about magnitudes as such in the absence of this relationship. With numbers the relationship of measuring is not essential to their definition so it is possible to consider both cases. The subsequent Definitions in Book VII provide a series of ways in which numbers can be viewed as magnitudes and structured by their parts. In Definitions 6–10 numbers are analysed into those with two equal parts (even numbers) and those which do not have two equal parts (odd numbers) and numbers measured by these. In Definitions 11–14 numbers are analysed as to whether or not they are measured by other numbers (primes and composites). Definitions 15–19 provide specific cases in which numbers are made up of two or three other numbers as parts (plane and solid, square and cube). Definitions 20 and 21 then extend a definition of proportionality to numbers. The difference in approach between Books V and VII is thus highlighted in that, whereas ratio and proportion are the concepts through which other definitions are stated for magnitudes in Book V, in Book VII no definition of ratio for numbers is given and the definition of proportion is in terms of multiples, part and parts, i.e. in terms of other, more basic concepts. ‘Part’ is the general context for magnitude and is therefore not available for further specification, while part and parts provide the ways in which numbers can be structured. The last Definition of Book VII concludes the process by defining perfect numbers as numbers which are completely constituted by their parts. Unlike magnitudes, numbers are constructed from their parts and their definition does not involve as its essence the relationship of one number to another. Numbers can be defined and dealt with individually and do not have to be dealt with in relation to other numbers. The characteristics of numbers, however, such as primality, parity and so on are precisely the relationships of measuring and being measured that they bear to other numbers. The theory of proportion is therefore important in the case of numbers in elucidating their characteristics, not in defining their nature. In all of this it must always be borne in mind that neither magnitudes nor numbers are ‘things in themselves’ but are only ways of regarding other things already known and defined. The remarks noted above concerning the lack of Postulates and Common Notions and the role of diagrams in demonstrations concerning magnitudes and their ratios and proportions carry over to numbers as well but with an important variation. Magnitudes are relative to the processes of measuring and being measured. Definition 3 of Book V (‘ratio’) places magnitudes in relationship to one another through the concept of ratio and specifies that the two magnitudes in the ratio must be of the ‘same kind’, i.e. the things they measure and which they are measured by must be the same. In Book VII by contrast, there is no mention of ‘kinds’ of numbers since

THE CONTEXTS OF MEASUREMENT 67

numbers are relative only to the unit. Units in turn are based on the existence of things which is the same for all things. Hence all numbers ‘have a ratio’ with respect to other numbers. There are two consequences of this. The first consequence is that there is no equivalent to Book VI following Books VII–IX on number; there is nothing for number to be ‘applied to’ in a similar way. The second is that propositions can take the form not only of demonstrations but also of ‘finding’ a number which satisfies certain requirements (e.g. finding the least common multiple of a collection of numbers).18 These propositions are not constructions in the sense of Books I–IV as there are no Postulates on which to base such constructions and no material on which to operate. Numbers remain a way of looking at already defined things. However, because numbers are put together from units and can be analysed in terms of their parts and because there are no ‘kinds’ of numbers, it is possible for Euclid to talk of ‘finding’ a number which satisfies some requirement. The process of finding is an ‘algorithm’ in modern parlance and shows how the parts of the number can be found and the number put together from them. Euclid terminates these propositions with the remark ‘QED’ and clearly distinguishes them from constructions or problems in the earlier sense of the word. This locution ‘to find’ a number (or, in Book X, a magnitude) is found only in Books VII–X and not in the remainder of the Elements. In Book V Euclid’s Definitions paralleled those of Book I by defining a notion of ‘horos’ (‘term’, ‘boundary’ or ‘definition’) and defining various types of ratios and the relationships between them in terms of the ‘terms’ in the ratios. The definitions in Book VII, however, operate in terms of the parts of numbers and the ways in which numbers constitute each other. This is made possible by the fact that numbers are made up of units, a fact which in turn makes impossible the kinds of reciprocal relationships of measuring and being measured that allowed the Definitions of Book V to proceed and which guided the sequence of Propositions. The sequence of Propositions in Books VII– IX reflects these differences from Book V as well. 3.6 Numbers and their parts Propositions 1 and 2 of Book VII set out the well-known ‘Euclidean algorithm’ for finding the greatest common divisor (using modern terminology, Euclid employs the term ‘greatest common measure’) of two numbers. This procedure is based on the continuing alternating subtraction of one number from another until a number is produced which either measures (divides) the previous number in the sequence or is a unit. In the first case, the number produced is the greatest

68 FIGURES OF THOUGHT

common divisor of the two numbers (Proposition 2), in the second case the numbers are prime to each other (Proposition 1). The proof operates by demonstrating that any number which measures the two given numbers must also measure the number that results from this process. If the result of the process were to be a unit this would imply that the greatest common divisor of the two numbers divides the unit which is impossible (since the greatest common divisor is a number and is therefore made up of units), hence the two numbers have no common divisor but the unit. Proposition 3 extends the algorithm by showing how to determine the greatest common divisor of three numbers. Proposition 4 states the crucial property of numbers which underlies the remainder of the analysis: Any number is either a part or parts of a number, the less of the greater. It is this fact, which results from the structure of numbers as being put together of units, that indicates how Euclid can proceed to analyse numbers in terms of their parts rather than basing his analysis on ratios and proportions. Magnitudes have no structure apart from their relationships to other magnitudes (compare Proposition 25 of Book V with this Proposition). Numbers by contrast do have internal structure which is sufficient to determine them and to determine their relationships to other numbers. The proof relies on the commonality of the unit in the constitution of numbers and the ability to find common divisors of two numbers using Proposition 2. Propositions 5–14 of Book VII carry over various aspects of Book V into this new context. Equimultiples are demonstrated, for example, not only for the case ‘a is the same part of a
that b is of b
’ but also in the case ‘a is the same parts of a
that b is of b
’. Hence there is a doubling of the propositions as found in Book V. While the verbal formulations of the Propositions are similar between the two books, the significance of the Propositions alters with the shifting context. Propositions 15–39 treat numbers as magnitudes and determine their parts and the ways in which they can be parts of other numbers. To analyse this a further gloss is required on the Definitions of Book VII. Euclid provides two Definitions which appear to a modern eye to be simple inversions of one another (particularly if they are written in algebraic notation) but which in fact require careful distinguishing. As discussed above, Definition 3 states that a number is a part of another number, the lesser of the greater, if it measures it, while Definition 15 states that: A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other.

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Modern eyes immediately see Definition 3 as ‘division’ and Definition 15 as ‘multiplication’. But a further distinction is required. As noted above, Definition 3 applies the context of magnitude, ‘part’, to numbers and carries over Definition 1 of Book V verbatim. Definition 15 on the other hand treats the two numbers in terms of their basic constituents, their units. At the risk of entitizing magnitudes and numbers one might say that Definition 3 treats numbers as magnitudes and Definition 15 treats them as numbers. Euclid feels no need to entitize either numbers or magnitudes, however. At various points in Book VII commentators have been puzzled by what appears to be a duplication in propositions.19 For example, Proposition 36 requires finding the least number which three given numbers measure (least common multiple) while Proposition 39 requires finding the least number with three given parts. The demonstrations of these two Propositions are hardly distinguishable and modern notation renders their statement identical. Euclid’s method, however, distinguishes the two in that the first treats numbers as magnitudes while the second does not. As nice as this distinction may appear at first it will be of great importance in Book X. These two points of view are brought together in Propositions 37 and 38 which may also ring oddly in modern ears: • if a number is measured by a number, the number which is measured will have a part called by the same name as the measuring number; and • if a number have any part whatever, it will be measured by a number called by the same name as the part. For example, since 3 measures 6, 6 has a part which is a third (i.e. 2); since 6 has a part which is a third it is measured by 3. The demonstration relates the concept of measuring to that of multiplying in that if b measures a then there is a number c such that c multiplied by b (i.e. c taken b times) is equal to a. It is by using this linkage that Euclid goes from the least common multiple of a collection of numbers (Proposition 36) to finding a number with given parts (Proposition 39)—i.e. from numbers as magnitudes to numbers as numbers. These Propositions show how Euclid is moving towards specifying the aspect of measurement left open in Definition 1 of Book V. There he had no way of specifying how many times one part measured another magnitude. Propositions 36–39 fully determine this in the case of numbers showing all of the relationships between the notions of ‘part’ and ‘measure’. The exploration of this for magnitudes will involve the notions of commensurable/incommensurable.

70 FIGURES OF THOUGHT

A complete review of Euclid’s procedure in the remainder of Books VII–IX would take us too far afield. The reader will note that it is characteristic of Euclid’s approach to number and its emphasis on numbers and their parts that he terminates this sequence of books with a method of generating perfect numbers, i.e. numbers which are equal to their parts both as magnitudes and as numbers. 3.7 Commensurable and incommensurable, rational and irrational Book X of the Elements is one of the great achievements of ancient mathematics and stands to this day as a testimony to the explanatory and exploratory power of mathematics. The legends20 surrounding the discovery of incommensurability by Pythagoras and his followers are well known. The success of the Greeks in using a mathematics based on reasoning rather than observation to obtain the startling result that two magnitudes are not always susceptible of being measured by the same magnitude is rightly considered one of the milestones in the development of science. It remains here to relate Euclid’s procedure in this book with that employed in Books V and VII–IX to complete this portion of the commentary on the Elements. The Definitions for Book X operate differently from those in any of the previous books. In the first place they are purely verbal, the operative phrases being ‘are said to be’, and they introduce neither new material nor new contexts in the sense of Book V or Book VII, but rather a comparison of the contexts introduced in these books. One might say that the definitions of Book X introduce a ‘second-order’ context. In the second place, Definitions 3 and 4 refer to the ‘hypothesis’21 that there are commensurable and incommensurable lines and go on to label the original line or square by which other magnitudes can or cannot be measured as ‘rational’ together with those magnitudes which are measured by it, all others being referred to as ‘irrational’. The reader must temporarily forget the meanings currently applied to these terms. The relationship of commensurability is relative to a given magnitude (line or square) and therefore the labels of ‘rational’ and ‘irrational’ are as well. Euclid specifies lines and squares as the magnitudes to be examined because he will now go about using the constructions in Books I–IV and Book VI to effect his demonstrations. The distinction between QED and QEF is not maintained however. The phrase ‘to find’, now applied to magnitudes not numbers, is carried over from Books VII–IX. The lines and squares are examined not as lines and squares but as magnitudes. The specific operations of ‘constructing’, ‘dividing’ or ‘circumscribing’

THE CONTEXTS OF MEASUREMENT 71

are relevant only in so far as they allow conclusions to be drawn about the relationships between magnitudes, their ratios and proportions. Books V and VII–IX provide two contexts in which the previously defined materials of Books I–IV can be viewed. The first is the context of magnitude, of measuring and being measured, which generalizes the relationships between geometric figures, particularly the relationship of equality, and views them as types of magnitudes depending on the kinds of things that each measures and is measured by. The second is the context of numbers where each thing is viewed simply as an existing thing, a unit, and numbers are put together from these. Although numbers do not have types or kinds they can alternately be viewed as made up of units or as magnitudes relative to other magnitudes. This dual functioning allows a complete specification of the concept of measurement for it specifies not only the fact of one magnitude measuring another but how many times it is measured by the other. This is the answer to one of the two issues left open by the treatment of measurement in Book V. Euclid now examines the other open question, when does one magnitude measure another? Book X explores the extent to which the two contexts of number and magnitude can be brought together, specifically it asks the extent to which magnitudes are susceptible to common measurement as are numbers, i.e. the extent to which magnitudes can be structured from parts.22 To do this Euclid takes the characteristic aspect of magnitude, the relationship of ‘measures/is-measured-by’ and applies it to two magnitudes simultaneously. Magnitudes in Book V were dealt with in proportions. Two magnitudes were never directly compared but rather their relationship was compared with the relationship between two other magnitudes. This always required dealing with more than two magnitudes at a time. Book VII opened, by contrast, with Propositions relating two numbers in terms of their common divisors. The opening Propositions of Book X resemble the opening Propositions of Book VII, but with a difference. Proposition 2 of Book X applies the algorithm of Propositions 1 and 2 of Book VII to magnitudes. Whereas the result of this process in the case of numbers was the conclusion that the numbers were prime to each other under the assumption that the result of the algorithm was a series of numbers none of which measured its predecessor, here the result is that the two magnitudes are incommensurable, that they have no common divisor or measure. The argument is similar to that employed in the earlier book, that a common measure of the two magnitudes must measure the remainder. In this case the result is proved by contradiction in that if the magnitudes were not incommensurable one could then prove that a greater magnitude would measure a lesser (in Book VII the contradiction relied on the nature of

72 FIGURES OF THOUGHT

the unit as a number not measured by another number). Proposition 3 takes the other alternative and shows how the process issues in a common measure if the magnitudes are commensurable and Proposition 4 extends the result to three magnitudes. Formally therefore there is a parallel between Propositions 1–3 of Book VII and Propositions 2–4 of Book X. The difference, both in the sequence of presentation and in the results, is Proposition 1 of Book X. Here Euclid proves the proposition that if a process of subtraction is started with a magnitude, from which a magnitude greater than its half is subtracted, and similarly with the remainder of this subtraction a magnitude greater than its half is subtracted and so on, the process will eventually produce a magnitude which is smaller than any given magnitude. Such a process if applied to numbers will produce parts of numbers, but not arbitrarily small numbers since all numbers are made up of units. There is and can be no equivalent to Proposition 1 of Book X in Book VII. It is this simple Proposition which, as Heath notes,23 justifies the use of the alternating subtraction algorithm in Proposition 2 and the conclusion in that case that the magnitudes are not commensurable. Proposition 1 of Book X shows the overlapping of the contexts of magnitude and number, part as measure and unit as building block. It relates two magnitudes as though they were numbers (i.e. without looking to proportional relations) and shows the extent to which they can and cannot be viewed as constituting and measuring each other. This inaugurates the new phase of the argument to be found in Book X. We are now in a position to respond to the standard criticism of Book X and the virtual consensus among commentators that Euclid’s exposition is flawed after Proposition 4.24 Proposition 5 asserts that commensurable magnitudes have to each other a ratio which is the same as the ratio of some two numbers. The purported problem in the demonstration is that Euclid has not, so it is said, shown how to relate the two definitions of proportionality given in Books V (Definition 5) and VII (Definition 21). Under the view put forward here, however, Euclid has been treating numbers as magnitudes since Definition 3 of Book VII and has shown in Propositions 37 and 38 of the same book that there is an equivalence between the concepts of ‘a measures b’ (number as magnitude) and ‘a is the product of a multiplication involving b’ (number as number). Simson stated correctly many years ago that Euclid nowhere sets out a proposition showing that if a:b::c:d as numbers then a:b::c:d as magnitudes. The lack of such a proposition does not mean, however, that Euclid has failed to provide a linkage between the two notions of proportionality. Furthermore, Simson’s approach, as discussed above, involves introducing certain axioms which entitize magnitudes and numbers to a far greater extent than

THE CONTEXTS OF MEASUREMENT 73

Euclid ever does without providing information over and above that given by Propositions 37 and 38 of Book VII. In Propositions 5–9 of Book X Euclid is able to characterize commensurable magnitudes as those magnitudes which have to each other the same ratio as a number has to a number (and commensurable squares as having the ratio to each other of square numbers). Here the power of the double treatment of magnitude and number is displayed. Ratios are relationships between magnitudes. Numbers can be considered as magnitudes when they are regarded as measuring and being measured and can therefore have ratios. But the ratios between numbers can be analysed in terms of their characteristics as numbers, their parts and their definition in terms of units. Hence particular ratios of magnitudes can be characterized as being the same ratios as two numbers and it is this characteristic which in turn decides their commensurability. The commensurability/incommensurability distinction is Euclid’s response to the second aspect of the indeterminacy of the relationship of ‘measures’ as laid out in Book V. Whereas in Book V Euclid is unable to determine whether one magnitude actually measures another, with Propositions 5–9 of Book X not only can the general question of commensurability be addressed, but specific cases can be set out as well. Since commensurable/incommensurable is a symmetric relationship, unlike the ‘bi-directional’ ‘part/measures’, Euclid is able to set out a rich collection of circumstances under which relations of commensurability and incommensurability can be observed. The double treatment of proportion when combined with the constructions of Books I—IV provides Euclid with very powerful tools to discern types of relationships between magnitudes and thus to relate lines and squares to one another in very subtle ways. Book X carries out a systematic programme of characterizing these relationships. The further definitions given after Propositions 47 and 84 (another example of the differences between the Definitions in Book X and those in the remainder of the treatise) provide a catalogue of various types of irrational relationship that can obtain between lines and squares. At no point are these relationships tied down to a fixed unit or a given real number system, however; they are all relative to whatever ‘rational’ has been originally selected (the ‘hypothesis’ of Definitions 3 and 4). The relationships of irrationality will be of use in analysing the parts of the figures which arise in Book XIII when the various parts of the solids constructed there are compared with each other and with the spheres which circumscribe them. It should be clear, however, that modern views of number as sequence and the ‘number line’ with its rationals and irrationals are irrelevant to Euclid’s approach. He is indifferent to the overall ‘structure’ of the ‘collections’ of magnitudes

74 FIGURES OF THOUGHT

and numbers because they are not entities in themselves but ways of looking at and studying other things. With this outline of Books V–X in place, the work of later mathematicians who grappled with concepts related to Euclid’s magnitudes and numbers can be reviewed. Few mathematicians have followed Euclid’s methods of placing previously defined geometrical materials in shifting contexts of measurement and part as a means of defining magnitude and number. Nonetheless the above analysis of Euclid’s use of ‘context’ provides very useful tools, for example, in reviewing the development of number and related concepts in nineteenth-century mathematics where the consequences of viewing number and ratio as things in themselves will become clear.

4 Number theory in the nineteenth century

4.1 A shift in subject matter From the end of the Renaissance through the eighteenth century European mathematics was dominated by geometry and allied fields. The work of Descartes in the seventeenth century had bound algebra together with geometry and the subsequent development of the infinitesimal calculus by Newton and Leibniz led to a greatly expanded power to examine geometric phenomena. The domination of geometry was so complete that the term ‘geometer’ was synonymous with mathematician. Beginning with C.F.Gauss at the turn of the nineteenth century all of this changed. His famous (and, for his contemporaries, almost impenetrable) work on number theory, Disquisitiones Arithmeticae, had provided a proof of the mysterious law of ‘quadratic reciprocity’ and had also provided the first new construction of a regular polygon since Euclid’s Book IV. Furthermore, Gauss saw the utility of using number systems involving ‘modular arithmetic’ and ‘imaginary numbers’. By the middle of the nineteenth century new kinds of number and number systems were regularly introduced by mathematicians, frequently with much fanfare and, occasionally, with some success. As we shall see, the end of the century took a more critical and reflective view, but the fact that after Gauss the ‘balance of power’ in mathematics had begun to shift from geometry to numbers is beyond dispute.1 Behind the mathematical issues which will occupy the remainder of this chapter there are important philosophical considerations which help to explain the shift away from geometry to the study of ‘number’ or ‘quantity’. Starting with the work of Immanuel Kant, philosophers routinely distinguished between the ‘intuitive’ and the ‘discursive’ in thought. Here ‘intuitive’ does not refer, as it might today, to vague and mysterious sixth senses and the like, but simply to sense perception (‘anschauung’).2 Kantian philosophy distinguished between the forms

76 NUMBER THEORY IN THE NINETEENTH CENTURY

of intuition (in his philosophy ‘space and time’) and the categories of thought and allocated intuition, by definition, to the ‘outside world’. Mathematicians, eager to ground their science in thought and not to depend on data from ‘outside’, sought to exclude intuition and to rely solely on logic and calculation. Gauss stated his views on this in a letter to Bessel: According to my innermost reflections Geometry has a very different place in our knowledge a priori, than the pure theory of quantity (or magnitude)
We must admit, with some humility, that while ‘Number’ is a pure product of our mind (‘Geist’), ‘Space’ has a reality outside of our minds and that therefore we cannot wholly prescribe its laws (‘Gesetze’) a priori.3 Gauss’ enormous influence and prestige ensured that this view became the common coin of mathematicians (at least in Germany), even to the extent that it was tacitly assumed rather than explicitly stated. The lines of development which emanate from this Gaussian position include Riemann’s famous Habilitationsvortrag on the ‘Hypotheses which lie at the Foundations of Geometry’ (the topic having been chosen by Gauss himself) and the large-scale enterprise which became known as the ‘arithmetization’ of mathematics. I shall be concerned with this latter line of development, and specifically with the Theory of Algebraic Numbers as developed by Dedekind and Kronecker. The breadth of the ‘arithmetization’ enterprise must not be underestimated. Mathematicians from Weierstrass to Cantor took up this cause and it led directly to Hilbert’s programme and the other ‘foundational programs’ that were such a feature of early twentieth-century mathematics. The legacy of this philosophical programme is still very much with us. Although the following discussion of Dedekind and Kronecker will leave these philosophical issues largely unexplored, they remain the most enlightening context in which the investigations of the concept of number undertaken by these two mathematicians may be placed.4 4.2 Kronecker, Dedekind and their predecessors Kronecker was slightly older than Dedekind, but the two men’s active careers largely overlapped as Kronecker spent much of the decade from 1845–1855 looking after his family’s business affairs. Dedekind was among the last mathematicians to have studied at Göttingen with Gauss and he felt the connection with his forebears very powerfully from the outset of his career. Kronecker was less directly connected to Gauss and

FIGURES OF THOUGHT 77

Göttingen but maintained close links of a scientific and personal nature with many of the professors there. In particular both men shared an enormous respect and affection for P.G. Lejeune-Dirichlet and both participated in the editing of his papers after his early death in 1859. The other mathematician with whom they both were linked was E. Eduard Kummer, who had been Kronecker’s thesis advisor and to whose work Dedekind devoted much of his time in the period from 1855 to 1875 (albeit without Kummer’s approbation). A grasp of the efforts of these two men is necessary for an appreciation of the thought of Kronecker and Dedekind. As always the story begins with Gauss. Gauss found the law of quadratic reciprocity so mysterious and profound that he provided no less than eight independent proofs of its validity. These proofs contain in embryonic form much of the mathematical development of the nineteenth century, including notions of theta functions and cyclotomic fields. Gauss went further and demonstrated a law of bi-quadratic reciprocity as well, thus beginning a mania for ‘reciprocity laws’ that has yet to subside. In all of this one of the most powerful notions that Gauss brought to bear on these problems was that of ‘complex numbers’, meaning numbers of the form a+bi, where i=√(−1) and a and b are ordinary integers (I shall use the technical expression ‘rational integers’ instead of ‘ordinary integers’ Subsequently, from now on to avoid confusion and shall write Kummer would study numbers of the form a+bµ where µn=1 is an nth root of unity. Other roots of positive and negative rational integers were substituted for i as well. It was quickly noticed that certain of these number systems did not obey some of the basic rules of arithmetic that Gauss himself had been the first to prove rigorously. The most important of these rules is that of ‘unique factorization into primes’. This states that any rational integer can be factored uniquely as a product of powers of prime numbers (e.g. 12=22×3). Simple examples such as 21=3×7 = [1 +2√(−5)] [1 −2√(−5)] show that even rational integers may not obey the law of unique factorization into primes when looked at as part of these larger number systems. This naturally raises the question of how to define ‘prime numbers’ within number systems of the above forms. In making such definitions one wants to maintain the property of ‘unique factorization’ so that all other numbers can be uniquely factored into powers of these new primes. Kummer discovered to his dismay and under circumstances which gave rise to some embarrassment that numbers of the form a+bµ where µ23=1 can satisfy no such requirements.5 The comfortable and useful rules of factorization into primes are foregone in the generalization of the concept of number to these new domains.

78 NUMBER THEORY IN THE NINETEENTH CENTURY

In a series of papers from 1845 to 1859 Kummer sought to rectify this problem by finding new ways to keep track of factorization and divisibility in his new number domains. To do so he introduced a new element, the so-called ‘ideal numbers’ (discussed in greater detail below), through which he was able to re-establish a new kind of unique factorization.6 With these new numbers in hand he made a thorough study of numbers of the form a+bµ, µn=1, which have become known as cyclotomic numbers (through the connection to ‘circle cutting’, i.e. the construction of regular polygons in circles—Book IV of the Elements remains in view). The other strand in the development prior to Dedekind and Kronecker is provided by Dirichlet.7 His lectures on Number Theory delivered to students at Göttingen were edited by Dedekind after Dirichlet’s death and became Dedekind’s chosen vehicle for further developing his thoughts on number theory. These lectures are marked by three features: • rigorous development of the notions of factorization and divisibility (essentially following Gauss’s lead) and showing their failure in the context of more general number systems; • a new exposition of the theory of quadratic forms and, more particularly, the theory of the composition of such forms and their analysis into ‘genera’ (Gauss had provided an exposition of this theory in his Disquisitiones Arithmeticae which few mathematicians, then or since, have been able to penetrate— Dirichlet made this subject available to the broader mathematical community); • new applications of tools from mathematical analysis to the theory of numbers, in particular, proof of the ‘Theorem on Primes in Arithmetic Progressions’ which remains one of the cornerstones of Number Theory. The originality of this work and the impact it had in making Number Theory one of the principal mathematical disciplines were enormous. Both Dedekind and Kronecker sought to develop the legacy inherited from Dirichlet, each in his own way. While Dedekind sought to improve upon Kummer’s work at various points, it is noteworthy that neither he nor Kronecker ever seems to have found occasion to suggest that Dirichlet’s efforts could be improved. It has become a widespread view that Dedekind’s and Kronecker’s contribution at this point, following the work of Kummer and Dirichlet, was the introduction of the general concept of ‘algebraic integer’.8 While there is no disputing the vital importance of this new concept, it is perhaps allowing history to be told from the point of view of its outcome

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to insist on this step in the first instance. The facts are that Dedekind himself spoke of the struggles he had in developing his ideas, that Kummer (among others) did not see the value of Dedekind’s efforts and that Dedekind’s approach differed greatly from Kronecker’s. It is perhaps only in retrospect that the development of a general notion of algebraic integers detaches itself clearly and seems an inevitable next step in the story. A less tendentious view of this stage in the historical development would be to say that Dedekind and Kronecker both realized that by setting the work of their predecessors in new contexts distinctions could be established between ‘integral’ and ‘fractional’ numbers replicating the familiar distinction between rational integers and fractions, thus opening up the possibility of extending familiar concepts such as factorization and divisibility from arithmetic to these more complicated settings. The contexts adopted by the two men to establish these distinctions were very different and the results obtained differed accordingly. Hilbert’s Zahlbericht, written at the end of the century, marks the first point in the historical development that a mathematician could calmly begin a treatise on Number Theory with definitions of ‘algebraic number’ and ‘algebraic integer’ without entering into a range of controversies. It should be noted that the strictly mathematical and number theoretic facts which are involved in the history were (more or less completely) known to the two protagonists at a fairly early date. By 1862 for Kronecker and 1871 for Dedekind (at the latest) the facts about ramification and discriminants discussed below were already understood. The subsequent twenty-year period was one of reflection and (re)formulation as both men developed their views as to the meaning of the facts they had discovered. In recounting the story I have tried to preserve a sense of the fluidity of the ideas involved prior to their crystallization at the hands of Hilbert (and others). To examine this history in detail I shall begin with a discussion of Dedekind’s development of number theory. The clarity of his writing and the care he took in its presentation make it easier to lay out than Kronecker’s which is notoriously turgid as he rarely found the time to provide a complete exposition of his results. Dedekind’s work has also been the subject of work by a number of historians in the last 25 years (most notably by H.M.Edwards whose influence will be visible throughout the following discussion) simplifying the task of setting out his ideas.9

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4.3 Dedekind’s mathematical method At the outset of his joint paper with H.Weber, ‘Theorie der algebraischen Funktionen einer Verändlicher’, Dedekind states that the objectives of the work are to present the chief results of Riemann’s work in function theory in a manner which is ‘simple, rigorous and fully general’.10 While these objectives may seem commonplace to the modern mathematical reader, it is instructive to see precisely what Dedekind meant by each of these three requirements. The word ‘simple’ for Dedekind refers to a mathematical presentation which deals directly with the underlying mathematical ideas and does not rely upon (what he perceives to be) the ‘accidents’ of mathematical formulation and expression. In ‘Bourbaki’ terms one might say that Dedekind defines mathematics in terms of underlying structures rather than their symbolic formulations. Dedekind of course speaks in the nineteenth-century language of ‘ideas’ and ‘laws of thought’ and not in the twentieth-century language of symbol systems and structures. Much of Dedekind’s soul-searching for a formulation of algebraic number theory can only be understood as a continuing quest to locate the ideas that lay behind the various formulations of the subject with which he was familiar. The word ‘rigorous’ for Dedekind refers to mathematical argument which does not bring in matter from ‘intuition’, i.e. from the senses or the forms of space and time, but which relies solely on the mind and its products for its cogency. In this sense, although he differed widely from other ‘arithmetizers’ such as Kronecker and Weierstrass, Dedekind too wished to arithmetize mathematics, since only arithmetic (which he viewed as a part of logic) relies solely on the laws of thought for its development. In his book Was sind und was sollen die Zahlen,11 Dedekind gave a full-scale reduction of the basic principles of arithmetic to logic and the laws of thought, providing one of the first examples of the enterprise which has come to be known as the ‘Foundations of Mathematics’. The motivation to show mathematics to be independent of ‘external’ determinations, a ‘free product of human thought’, runs throughout Dedekind’s work and is particularly important in his number theory. Finally, the phrase ‘completely general’ for Dedekind means that all instances of a phenomenon are to be treated together, under one common definition. In his book on rational and irrational numbers, Stetigkeit und irrationalen Zahlen,12 Dedekind provided a definition of ‘real numbers’ based on collections of rational numbers which encompassed all types of ‘irrationality’. In a later paper he pointed to this definition as being vastly superior to others which defined numbers

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through various means (rational numbers, algebraic numbers, transcendental numbers of various sorts etc.).13 The drive to create definitions which ‘admitted of no exceptions’ is the third of Dedekind’s motivations that will be seen to operate in his number theoretical explorations. Already in his Habilitationsvortrag of 1854,14 Dedekind had spoken on the introduction of new concepts into mathematics (and other sciences) and had stated that in mathematics the purpose of introducing new concepts (he instances negative numbers, non-integer exponents and circular functions in his discourse which ends with the problems of ‘integrating’ elliptic and abelian differentials) is to complete the systems with which mathematics deals so that the rules governing the various operations are valid in as large a realm as possible. Dedekind remained faithful to this notion to the end of his life, seeking out general definitions that extended already known laws or rules to larger domains. His first major effort in this connection was the discovery of the definition of a ‘field’ (defined loosely as a collection of numbers closed under the four basic arithmetic operations of addition, subtraction, multiplication and division). There is no certain date for this discovery, although recent work on Dedekind’s early lectures on algebra suggest that it was during the period 1855–1860, well before his major work on number theory.15 It may be that the thought process set out in the Habilitationsvortrag led him directly to this point. In any event there is no mistaking the importance of this definition nor the cardinal place it occupied in Dedekind’s work. He states that he chose the name ‘Körper’16 because of its use in many sciences (including the biological sciences) in referring to things that are ‘complete, entire and closed’.17 I shall show the importance of viewing fields as the context in which Dedekind develops his number theory and, in particular, the context in which he locates the distinctions between integral and fractional elements. 4.4 Dedekind’s struggle towards algebraic number theory. Phase I: the discovery of ramification The principal vehicle for conveying Dedekind’s thought on number theory was the series of editions of Dirichlet’s lectures on number theory which Dedekind edited (to be referred to therefore as Dirichlet— Dedekind) and to which he appended a series of supplements. The first edition dated from 1863 and included a series of supplements on various topics. The second edition, in 1871, included a tenth supplement which presented Dedekind’s ‘ldeal Theory’ for the first time, in the guise of treating the composition of binary quadratic forms,

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the subject that Gauss had revolutionized and Dirichlet had reformulated and presented to the mathematical community. In 1879 and 1894 Dedekind published third and fourth editions, most of the changes being in the mode of exposition of the Ideal Theory which went through virtually a complete metamorphosis18 (along the way the tenth supplement became the eleventh supplement). The 1894 edition became canonical, although Dedekind apparently continued to work on revisions for a fifth edition until the end of his life. A certain amount of historical work has already been undertaken in tracing the development of Dedekind’s thought through these editions and the results of these historical enquiries will be incorporated in what follows. Most historical study to date has focused on Dedekind’s development of ‘ideal theory’ as a means of re-establishing the law of unique prime factorization without requiring the introduction of new entities such as Kummer’s ‘ideal numbers’.19 The review which follows therefore considers other aspects of this history related to the birth of (what has become known as) ‘class field theory’.20 The development of Dedekind’s Number Theory can be divided into four phases. Of the first of these phases the only published remnant is an early article on ‘higher congruences’ and certain remarks of Dedekind’s in later writings which allow us to reconstruct his thinking from this period. The later phases are associated with the second, third and fourth editions of Dirichlet—Dedekind respectively. I begin with the first phase. One of Dedekind’s earliest published papers was entitled ‘Abriss einer Theorie der höheren Kongruenzen in bezug auf einen reellen Primzahl-Modulus’21 (‘Sketch of a Theory of Higher Congruences in Relation to a Prime Number’, to be referred to here as ‘Abriss’). This paper lays out the theory of factorization of polynomials with rational integer coefflcients when they are viewed modulo a prime number or powers of a prime (in modern terminology this is the study of the reduction of polynomials modulo prime powers and, although Dedekind does not pursue this, in the limit, over the p-adic numbers).22 He develops a notion of ‘prime function’, i.e. a polynomial which cannot be factored (is ‘irreducible’) modulo a prime power, and explores the factorization of polynomials into these prime functions modulo powers of rational primes. The background for this work resides in Kummer’s analysis of factorization in his cyclotomic numbers. The example of 21 considered as part of the number system since has already indicated that a rational integer may not exhibit unique prime factorization when considered as an element in a larger number system. Nonetheless Kummer’s great success had been his

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ability to show that the factorization of a rational prime when looked at as part of one of the systems of cyclotomic numbers (i.e. its factorization into ideal prime factors) was entirely governed by the algebraic equations that these numbers satisfied. Specifically the cyclotomic numbers are created by adjoining a so-called primitive nth then we consider where root of unity to the integers (if in a manner wholly analogous to the way Gauss had adjoined i (a primitive fourth root of unity) to create his numbers a+bi. Kummer (and others) had proved that these numbers satisfy specific algebraic (monic polynomial) equations of degree n or lower (so-called cyclotomic equations). Suppose that one is given a cyclotomic number and consider the β which satisfies a cyclotomic equation where a and b subsequent collection of cyclotomic numbers as are integers.23 If we want to know how a rational prime p factors as an ideal in this system of cyclotomic numbers24 then we need only ‘reduce (mod p). F modulo p’ i.e. we need only study the congruence Now F(x) as a polynomial factors modulo p into a product of powers of irreducible polynomials Pi(x) so that (mod p) (note that the polynomials Pi may have degree higher than 1). Kummer was able to show that, as a cyclotomic number, p had a factorization into ideal prime factors which was patterned precisely after the factorization of F(x) modulo p (in fact, in the cases that Kummer considered, the factorization of F modulo p was simplified in that j=k==q).25 Dedekind’s intention was to extend Kummer’s approach to other number fields (something which, according to Kummer, Kronecker was meant to be doing at the same time).26 It seems likely that Dedekind had already developed his general notion of field and number field and he therefore sought to explore this new broad context with the tools that had worked so successfully in previous cases. This was not to be. To explain why, I refer to the notice that Dedekind wrote in 1871 at the time that the second edition of Dirichlet—Dedekind (containing the first published version of his ideal theory) was published.27 Kummer’s cyclotomic numbers display a number of very pleasant properties which make them easily accessible to detailed mathematical study. Symptomatic of these properties is the existence of a so-called power basis. If µ is a primitive nth root of unity, then all cyclotomic numbers created by adjoining µ to the rational integers can be expressed in a unique manner as linear combinations (with rational integer coefficients) of the powers of µ up to and including the (n−1)th power. Specifically, if β is a cyclotomic number then where the Unfortunately, in general (the rings of integers in) number fields do not have such generators. As a result, the factorization of rational primes when

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considered as elements of these fields does not depend solely on the algebraic equations that the numbers of the field satisfy (as was the case with cyclotomic numbers), but on another factor which is much more difficult to control. If the number field does not have a power basis, i. e. if the numbers in the field cannot be expressed as linear combinations of one power of one number alone but rather require combinations of powers of two or more numbers, these more complex bases reflect in turn the very complex behaviour of the rational prime considered as an element in the number field. This behaviour may not be detected by the process sketched above of looking at the factorization of algebraic equations modulo p (Kummer had in fact run into this problem in studying certain extensions of the cyclotomic integers now known as ‘Kummer Fields’ and had proceeded by simply avoiding certain special cases).28 The effect is visible when one relates the so-called discriminant of a given algebraic number to the discriminant of its number field (see the appendix to this chapter for definitions and a brief discussion). These two discriminants are linked by a number which is called the ‘index’ of the given algebraic number. Dedekind was able to show fairly easily that Kummer’s approach could be carried over to more general algebraic number fields if and only if for each rational prime p one could find an algebraic number in the number field whose index was not divisible by p. He recounts the story on more than one occasion of the effort that he put into demonstrating that such an algebraic number must exist in any number field for each prime p. After many years of failure it gradually dawned on him that the statement he had been trying so hard to prove might be false! Efforts in this direction were rewarded with a counter-example,29 a field in which, for some rational prime p (in the case Dedekind discovered, which has become the standard example in all texts, the prime in question is 2), the index of every number was divisible by p (i.e. the index of every number is divisible by 2, see the appendix to this chapter). Although the particular problem Dedekind had come up against is now well understood, the causes which give rise to it still provide many mysteries to number theorists. In modern terms Dedekind had come up against one aspect of the phenomenon of ‘ramification’,30 one of the most subtle and pervasive concepts in number theory and related geometric subjects and one which is particularly complicated for number fields. As I shall show, although Dedekind was able to prove some far-reaching theorems in this connection, he was never able to bring the topic to a satisfactory conclusion (bearing in mind his strict requirements for generality as noted above). The result is that, in spite of its importance, a general discussion of this topic was never included in the various editions of Dirichlet—Dedekind. Nonetheless the problems of ramification remain in the background like a brooding

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presence and it would be an error to read the various versions of Dedekind’s ideal theory without bearing it in mind. Conversely, a knowledge of this problem provides the clue to much of Dedekind’s thought. The first result of this discovery was the abandonment of the train of thought in Phase 1. In his later writings, Dedekind refers to ‘Abriss’ only to show how his first approach met with failure. Other effects are matters of speculation. Did his failure in applying Kummer’s approach in one area make him suspicious of the solidity and generality of Kummer’s work in general and thereby lead to his programme of ideal theory? This seems a natural conclusion, but one that must remain unverified (and probably unverifiable). 4.5 Phase 2: first introduction of fields and ideals Phase 2 of Dedekind’s evolution is represented by the second edition of Dirichlet—Dedekind in which the theory of ideals is laid out for the first time. It would be helpful to know at what point Dedekind became fully cognizant of the problems with his Phase 1 approach and, in particular, at what point he discovered his counterexample. It seems unlikely, however, that the data required to establish the chronology of these discoveries will ever be unearthed. Setting aside the psychology of the matter, I shall show that the discovery of ramification in the context of general number fields gave impetus to looking both at fields and at ideals as subjects of interest in themselves and not merely, as Dedekind seems to suggest they might have been at first, as clearer and more ‘orthodox’ versions of Kummer’s ideal numbers. In the second phase of his thought, fields and ideals are at the forefront, although mixed with ideas taken from Kummer, suggesting that the discovery of ramification and the break with Kummer’s methods might still be relatively fresh. In her notes to Dedekind’s collected works, Emmy Noether (whose mathematical work followed Dedekind’s approach more closely than any other later mathematician’s) distinguishes the exposition of the second edition of Dirichlet—Dedekind from subsequent versions of the theory by referring to its use of ideas from the theory of ‘hyper-complex’ numbers31 (modern terminology would refer to the theory of ‘algebras’ over the complex numbers, or other base fields). In spite of the antiquated flavour of this terminology it is useful to enquire into the meaning of these remarks. Section 159 of the second edition is entitled ‘Endliche Körper’ and provides a definition of a field (which he calls ‘the foundation of higher algebra and the related parts of number theory’) and definitions of

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norms and discriminants in fields. These definitions are phrased in terms of the algebra structure of the field.32 That is, he represents and writes elements α of the field in terms of a basis xi, say where the Hi are functions of the αi. He then studies these functions Hi. The multiplicative structure of the field thus plays a crucial role in the analysis, Conjugate fields are defined as fields which (as we would say today) are isomorphic images of each other (as embedded in the complex numbers) and elements of the field are ‘conjugate’ to their isomorphic images. Norms and discriminants are defined in terms of the conjugates. Dedekind introduces the notion of algebraic integer in section 160 (thus, as he puts it, ‘moving closer to [his] proper subject’) as a number which satisfies an equation of the form F(x)=cmxm+cm−1xm−1++c0, where the cm−k are rational integers. He states that in all such equations we are to understand the coefficients cm-k as in fact being ratios (−l)kCi/Cm. In other words, cm can always be taken as 1. With this understanding, the generalization from rational integer to algebraic integer does not alter the older notions since the rational integers are the only rational numbers to satisfy the definition of algebraic integers.33 This is the only discussion of the definition of algebraic integer to appear in Dedekind’s published work! Kummer (and Gauss and Dirichlet before him) had dealt exclusively with algebraic integers (from this new point of view) and had not distinguished integral from fractional elements within such fields.34 From the outset Dedekind thought the notion of ‘field’ to be crucial to his mathematics and hence the integral/fractional distinction as well, but as we shall see, his view of the precise role played by these concepts altered over time. Dedekind shows that the algebraic integers of a number field form a ring (a collection closed under addition, subtraction and multiplication but not necessarily division) and he begins to develop basic notions of arithmetic with these new integers. Concepts such as ‘relatively prime’ are easily carried over from the rational integers, but he signals that the concept of prime number is not so easily treated. In section 161 he defines the notion of ‘module’ which will come to play a much more important role in the later expositions, but here is used only to carry over the notions of congruence that Gauss used to such effect in his number theory (in modern parlance, reduction modulo an ideal). In the next paragraph Dedekind extends the notion of discriminant previously defined to include any collection of ‘linearly independent’ numbers of the field. The notion of the field discriminant or ‘Grundzahl’ of the field is then introduced and Dedekind is careful to point out its significance (as well as to refer to papers of Kronecker’s dealing with the matter). This discriminant is defined as the discriminant of a basis of the field (which is by definition a collection of linearly independent

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numbers) which has the properties that (1) it is made up of algebraic integers, and (2) for all algebraic integers in the field their ‘co-ordinates’ in terms of this basis are also algebraic integers. That is, a basis xi made up of algebraic integers such that, for all algebraic integers θ in the field, where the θi are also algebraic integers. Although he does make some use of this notion in later sections of the second edition, the reader not familiar with Dedekind’s other number theory papers (one who was for instance unaware of the notice that Dedekind wrote in connection with the appearance of the second edition of Dirichlet—Dedekind) might be forgiven for thinking that his use of the concept of field discriminant here does not entirely justify his insistence on its importance. In carrying out the comparison of the divisibility properties of rational integers and algebraic integers, Dedekind notes the failure of the basic laws of factorization for algebraic integers and remarks on Kummer’s strategy of defining divisibility by ideal numbers as a way to re-establish the basic principles of arithmetic. He concludes with the following, Only the fear that a direct carrying over of the usual methods of dealing with rational integers might lead, at least at the outset, to misunderstandings that could mar the certitude of the demonstrations led me to investigate how the matter might be differently presented, by working only with systems of ‘existing’ instead of ‘ideal’ numbers.35 Do these words carry the memory of the failure of Dedekind’s earlier programme or do they merely reflect a more scrupulous attachment to definitions and foundations than Kummer exhibited? We shall likely never know. What is clear is that Dedekind is not claiming revolutionary status for his new ideas at this point in time. In section 163 he defines ideals as systems (modern terminology would be ‘sets’) of algebraic integers which are closed under addition and subtraction and which are such that the product of any algebraic integer and a number in the ideal is also in the ideal. An ideal A is said to divide an ideal C when C is contained in A. C is also said to be a multiple of A (the collection of all multiples of 6 is contained in the collection of all multiples of 3). While no definition of multiplication of ideals is provided, notions of ‘prime’ ideals (ideals not contained in other ideals) and the ‘least common multiple’ and the ‘greatest common divisor’ of two or more ideals can be defined. Because there is no definition of multiplication of ideals, certain definitions and characteristics are defined not in terms of the ideals themselves but in terms of congruences that the elements of the ideal satisfy. In effect Dedekind is merely clothing Kummer’s work in new terminology. The

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key definition is not that of ‘prime’ ideals but that of ‘simple’ ideals, which are defined in terms of congruences. The fundamental ‘factorization’ theorem states that any ideal is the least common multiple of the powers of prime ideals which divide it. This presentation of the foundations of number theory undergoes substantial revision in Phases 3 and 4. H.M.Edwards has raised the question as to Dedekind’s reasons for wishing to modify this approach, which in some respects may be viewed as more satisfactory (and certainly less prolix) than the later versions. He attributes to Dedekind a desire to expand the presentation to make it more easily readable (he was apparently disappointed with the reception of the second edition) as well as a growing desire to defend points of view he had taken in publications on other subjects that had appeared in the meantime (most notably his views on irrational numbers).36 Without necessarily contradicting any of Edwards’s conclusions I would like to show how the evolution of Dedekind’s thought can also be explained in the light of his mathematical method and approach. 4.6 The third phase of the evolution of Dedekind’s number theory In 1876, Dedekind produced a long article in French which appeared in Liouville’s Journal and which was designed to expose his new ideal theory to the French mathematical community.37 While few French mathematicians showed much interest in the work, it is useful to us in showing an intermediate stage in Dedekind’s thought between the second and third editions of Dirichlet—Dedekind. Here Dedekind tells for a second time the story of the success of Kummer’s work (not mentioning the points discussed above respecting the factorization of rational primes as cyclotomic numbers) and discusses his own efforts. The story, however, comes out a bit differently: The success of Kummer’s efforts in the cyclotomic domain gave reason to believe that the same laws subsisted in all numerical domains, even those of more general type than his. In my work, which was aimed at a definitive solution, I based myself on the theory of higher congruences, since I had previously noted that the application of this theory would allow a considerable simplification of Kummer’s work; I was however unable to treat certain exceptional cases by following this path. I only arrived at a general theory, one without exceptions, after I gave up my previous formal method and replaced it with another which proceeds from the simplest of basic concepts and aims directly at

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the objective. In this way I have no need for new creations such as Kummer’s ideal numbers, and it suffices to consider systems of ‘existing’ numbers which I call ‘ideals’. The power of this idea lies in its simplicity
38 The reader has undoubtedly noted the recurrence of Dedekind’s criteria of generality and simplicity in his recounting of this story. Although Dedekind’s review of Kummer’s work (just prior to the above quoted passage) omits the points on factorization of rational primes, and although the subsequent context of the French article only deals with ideals and their factorization, and not with the behaviour of rational primes, it seems inconceivable that Dedekind does not have reference in this autobiographical passage to the ramification problems noted above. Yet how do we go from the problems of factorization of rational primes in number fields to the general theory of ideals? It would seem that Dedekind has here conflated the two quite different issues of re-establishing the laws of prime factorization with the question of how rational primes factor when viewed as elements of rings of integers of algebraic number fields. To understand what has happened we need to bear in mind Dedekind’s overall philosophy. The consequence of the failure of the project begun in Phase 1 was not merely that Dedekind discovered that the behaviour of rational primes in number fields is subject to more complex rules than is the case with cyclotomic numbers. Dedekind had expected to formulate his entire theory of algebraic integers on the congruence properties set out in his ‘Abriss’ and utilized by Kummer. There would have been no need for a separate and self-standing theory of ideals (or number fields for that matter) had he succeeded. The congruences themselves would have done the job (and we shall see that this position was defended by Kronecker in spite of the problems noted above). Dedekind’s ideals would then have been useful merely in simplifying the presentation and avoiding the kinds of errors that had allowed him to think that what worked in the cyclotomic case would work more generally. From this point of view the great step forward that Dedekind needed to take was to divorce the general theory of ideals from the factorization of rational primes, thus creating a ‘fully general’ theory. Hence we might say that the introduction of the general concept of number field showed up the insufficiencies of his thinking in Phase 1 (by introducing fields which exhibited complex ramification behaviour) and that the second edition of Dirichlet—Dedekind indeed embodies a theory of ideals, which at least in some respects, is independent of the factorization of rational primes in number fields. However, in other respects Dedekind had not fully integrated the ideals into the theory. Recall in this connection the extent to which the presentation in the second edition relied on

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congruence conditions to define crucial concepts such as ‘simple’ ideal. This suggests that Dedekind had not yet fully grasped the import of his ideas. By the time of the French publication a subtle change had taken place. Dedekind realized that the failure of the first phase was not only a matter of the problems of ramification but that in fact his introduction of fields, rings and ideals as independent entities committed him to a completely different view of the subject. If the properties of the ideals could not be established by moving ‘upwards’ from the rational primes to their factors in the number field, the ideals themselves should be viewed as carrying the divisibility properties of the field quite independently of the behaviour of the rational primes. Thus as simple and direct a presentation as possible of these facts was required. Here we can see the result of Dedekind’s constant search for simplicity in his thought, namely for mathematical argument based on ideas and not the ‘accidental form’ in which a computation may present them; for it was precisely in paying attention to the form in which the prime ideals were represented that he had previously gone wrong. From this point on the argument must free itself from particular forms of representation, i.e. free itself of references to congruences and particular numbers. In Dedekind’s view of the subject in Phase 1, not only the rational primes but in fact the integers in the number field were ‘composed’ of ideal factors as well, hence when Dedekind later began to view the ideals as subsisting independently of the factorization of primes, he also began to see the integers in the fields and indeed the fields themselves as having a similar independent existence. This was indeed revolutionary, and it took Dedekind some time to appreciate fully the scope of what he had accomplished. In fact, as the French article was going to print, he wrote to a correspondent that even its presentation was in some respects ‘marred’ by the fact that the fields in question were not dealt with as independent entities but were rather shown as ‘extensions’ of the rational numbers ‘generated’ by roots of certain equations.39 The realization of the full import of his ideas was developing rapidly at this point. Ultimately, Dedekind’s ideal theory succeeded precisely by freeing itself from concern with the behaviour of specific primes. This should be viewed as a vindication of his deeply held belief in the power of mathematical concepts. Dedekind expressed all of this very clearly in another article from 1877,40 in which he showed the relationships between the older theory based on congruences and his newer theory of ideals. He reviews his slice of mathematical autobiography yet again, providing a third version of the story. He tells us that he had worked on Kummer’s theory for many years without publishing anything because of two flaws that he had identified in it.

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[The first] that the investigation of a domain of algebraic numbers which is founded on a given number and the equations it satisfies expressed as congruences, and the subsequent definition of ideal numbers (or more specifically, divisibility by an ideal number) does not derive from the particular form of representation employed the character of invariance that belongs to this concept in truth. Second, that certain exceptional cases arise requiring special treatment. My new theory in contrast is based exclusively on such simple concepts as field, integer and ideal whose definition requires no specific form of representation.41 The story, at least in outline, is now familiar. He no longer sees his fields as generated by numbers satisfying algebraic equations and hence is no longer concerned with the congruences to which such equations might give rise. The exceptions caused by ramification are no longer a problem for the basic definitions of the theory (although, as discussed below, there remained important questions to resolve in this connection). It is clear then, that by the mid-1870s Dedekind had a very clear idea of the extent to which he had broken with the past. From this point on his confidence in his methods grew steadily. I return to the analysis of the French presentation to note two additional elements introduced there which take on increasing importance in the later phases of Dedekind’s number theory. First is the emphasis on the need for a general definition of ideals, in particular one that will permit the introduction of a notion of multiplication of ideals. His growing conviction in the independent status of the ideals may have had something to do with introducing this idea, in any event it forms a new point of departure for the argumentation that he employs, particularly in the third edition of Dirichlet—Dedekind which will be discussed shortly. Second, in Dedekind’s French exposition, while he defines an ideal as he had done in the second edition, explaining that the collection of numbers which make up an ideal has the properties that a collection would have if it were the collection of multiples of some number, he goes on to say that after great pains he was able to prove that in fact the converse was true in number fields. That is, any collection of numbers satisfying the two requirements to be an ideal was in fact a collection of multiples of some number (the modern terminology is to call such a number the ‘generator’ of the ideal). The crucial point is that the number in question might not be an algebraic integer, it might be an ideal number. It is then one and the same to speak of an ideal in the ring of algebraic integers of a number field and to speak of the number which generates it.42 The failure of the laws of factorization and divisibility can then be attributed to the existence of ideals which are not generated

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by algebraic integers (such ideals do not exist in the case of the rational integers) and in fact the extent to which the laws fail can be measured by the number of essentially distinct ideals which are generated by ideal numbers (this is the so-called class number and relates to the Gauss-Dirichlet Theory of Quadratic Forms). In any event, the emphasis that he places on this converse theorem (together with his own remark about the use of generating elements to present the fields) shows that Dedekind at this point in his development had not completely freed himself from the temptation to relate ideals to numbers. However, this was to be the last vestige of this type of thinking in Dedekind’s development and by the time of the third edition (1879) the process of abstraction and separation was complete. In his treatment of the subject in the third edition of Dirichlet— Dedekind, the emphasis is on the notion of multiplication of ideals, which he here calls the ‘proper core of ideal theory’, and the objective of ideal theory is stated to be the study of the interaction between the notion of divisor/multiple based on the relationship of inclusion (as discussed in the second edition) and the notion of multiplication of ideals employed here (a clear echo of Euclid’s concerns in Definitions 3 and 15 of Book VII!). This formulation is also adhered to in the fourth edition. Noether notes that Dedekind has moved away from the ‘hyper-complex’ number approach of the second edition but has not yet adopted the Galois theoretic approach of the fourth. Conjugates are still defined in terms of isomorphic images of the field and discriminants and norms in terms of these. The new idea introduced is that of ‘module’ which provides the framework for the analysis of ideals. In section 166, having provided the necessary background, Dedekind introduces the ‘proper object of our investigation’, namely the distinction between fractional and integral elements in a field. This leads immediately to the definition of the field discriminant in terms of a set of generators for the ring of integers (not a basis for the field as in the second edition) and then to the factorization of ideals. The conclusion of the theory as presented here is that any ideal can be factored in a unique manner into a product of prime ideals and this factorization can be related to the factorization into the least common multiple of prime ideals shown in the second edition. 4.7 The final phase of Dedekind’s evolution In 1882, alongside the report on Algebraic Functions with Weber, Dedekind produced perhaps his greatest individual paper in number theory, ‘Uber die Diskriminanten endlicher Körper’ (‘On the Discriminants of Finite Fields’—today we would say Number Fields),43

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in which he discussed in detail the problems of ramification which had forced him to abandon his initial formulation of number theory. In this paper the concept of field discriminant is used with its full power and Dedekind is able to show that in almost all cases the field discriminant controls the behaviour of rational primes in number fields. The details of the paper are too technical for presentation here, but a few general points must be noted. First, the use of the field discriminant in revealing the behaviour of rational primes implies that this concept must have a central place in number theory. Paradoxically, for reasons which will become apparent shortly, the role of this concept in the general scheme of Dirichlet—Dedekind seems to diminish with each edition. It should be recalled that in the third edition Dedekind had proven that ideals have a factorization into prime ideals and powers of prime ideals. In this paper he demonstrates: • that those prime numbers and only those prime numbers which divide the field discriminant ‘ramify’, i.e. have a factorization as an ideal in the number field in which a prime ideal appears to a power higher than one (since the field discriminant is a rational integer44 it can only have a finite number of prime factors, hence only a finite number of rational primes ‘ramify’ in a given number field); and • in almost45 all cases there is a simple and direct relationship between the highest power of the prime number which divides the field discriminant and the highest power of a prime ideal figuring in the ideal factorization of the prime. In this paper Dedekind has almost entirely resolved both of the problems encountered in going from the arithmetic of the rational integers to the arithmetic of the integers of algebraic number fields. His ideal theory re-established a notion of ‘unique prime factorization’ and the notion of the discriminant enables him to say, at least in most circumstances, how the rational primes factorize when considered as algebraic integers. Note that these two issues are now logically separated into a universal fact (that of unique factorization of ideals) and facts which are specific with respect to particular number fields (the prime factorization of the discriminant). The factorization of ideals is ‘prior’ both because it is universal and because it gives rise to the factorization of the discriminant. While this paper contains some of the most profound and general statements ever proven about number fields, the existence of the few exceptional cases clearly kept Dedekind from accepting it as the last word on the subject. This also explains Dedekind’s steady reduction of the role played by the field discriminant in the various editions of Dirichlet—Dedekind. The fourth edition of the book does not even mention the topic of primes which divide the

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discriminant although, of course, the matter of the factorization of ideals is taken up in great detail. The field discriminant is only introduced in section 175, rather late in the argument and without much fanfare. Since Dedekind’s time, his theorem on primes which divide the discriminant has not been improved upon (at least taken in its full generality); instead number theory has advanced principally by looking at specific types of number fields and number field extensions. This is precisely what Dedekind himself would have wished to avoid! In retrospect we can see how the phenomena of ramification posed basic problems for the definition of ideal prime factors in Phase 1 of Dedekind’s evolution. Following his drive for generality and theories admitting no exceptions, he found he could not employ generalizations of Kummer’s methods to define ideals. Only after he had re-established the theory, based as he says on the simple definitions of fields, ideals and integers, could he revert to the problem of ramification. With ideals and their prime factorization defined and proved independently, the factorization of rational primes in number fields was no longer a problem of definition but rather a set of relationships between the rational numbers and the newly defined number fields as governed by the factorization of a particular ideal, the field discriminant. The fourth edition of Dirichlet—Dedekind differs principally from its predecessors in that the notion of field is introduced directly and the algebraic integers and fractional elements are located therein. The previous editions had begun, as had the mathematics, with the algebraic integers and the field only came in at a later stage. With the fourth edition Dedekind sets out his context from the beginning and proceeds directly to its study rather than mixing the study of the field with that of the ring of integers. The field can be studied first as a ‘linear’ object (i.e. without reference to the multiplication as in the second edition) over Q and this leads to the Galois theory of fields, i.e. the theory of linear transformations of number fields viewed as vector spaces over the base field. In this he followed the ideas of the paper of 1882 on discriminants. At the same time Dedekind had become aware of the power of his methods and was more confident of the grounds of his argument. The fourth edition shows almost no sign of congruence techniques nor even of the association of ideals with generating numbers which figured in the French version of the theory. Ideals are treated as a special case of modules, and a full treatment of ‘orders’ in rings of algebraic integers is also included. The key concept is no longer factorization of ideals into powers of prime ideals but the demonstration that the collection of ideals forms a ‘group’ under the operation of multiplication of ideals that had been introduced in the French version of the theory. The generality and ‘abstract’ quality of the

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presentation in the fourth edition can surprise even twentieth-century readers.46 The story told here shows how Dedekind pursued his goals of simplicity, generality and rigour throughout the development of his number theory and with increasing confidence in the power of the methods he was using. His chosen context in which to pursue this work was that of number fields. The breadth and scope of this new context forced him down a path in which a very general distinction between algebraic integers and ‘fractions’ was rendered both possible and necessary, whereas it had been both unknown and unnecessary for his predecessors. The ideals can be seen as ‘in’ the ring of integers of a field or as ‘in’ the field (as ‘fractional’ ideals) but the distinction between integer and fraction is one that is intrinsic to the field and cannot be carried over from the familiar distinction of integers and rational numbers due to the problem of ramification. At the same time the generality of his context and the intractable nature of ramification led him to develop a theory of ideals in number fields divorced from the easily representable congruences involving primes that his predecessor Kummer (and even some of Dedekind’s contemporaries) had tried to employ. The new context of number fields provided him with ‘room’ in which to manipulate his ideals and modules and the great variety of number fields provided him with deep problems to which he was not always able to find completely general solutions. The distinction between integers and fractions allowed him to develop analogies between the rational integers and algebraic integers using the ideals as a mediating point, but this was only possible once he had cut his ideals loose from their moorings by the side of the rational primes and allowed them to find their natural formation within number fields. Kummer never seems to have appreciated the efforts that Dedekind had gone through to develop his ideal numbers into a new theory. Certain remarks suggest that he may not have understood the extent to which Dedekind had resolved the problems of ramification of which Kummer had been aware in the 1850’s.47 In modern times the differences of method between Kummer and Dedekind (and later between Kummer’s student Kronecker and Dedekind) have been viewed as the first clash between two ‘philosophies’ of mathematics that have become known as ‘constructivist or intuitionist’ and ‘platonist’. This may be so. The interpretation presented here suggests that there may be another, perhaps more ‘mathematically practical’ reason for the difference of views. According to Dedekind, the number theoretic problems located by his methods resided in the contexts of his new number fields, not taken one by one or even class by class, but rather taken in their generality. They were problems which were only brought to light when this new context was introduced. From

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Dedekind’s point of view, to treat individual cases or types of number fields except as examples was literally to miss the point. To understand the other side of this argument I shall turn to the work of Leopold Kronecker, Kummer’s student and great friend, whose views on mathematics and mathematical method could not have differed more profoundly from Dedekind’s. It should come as no great surprise to learn that Kronecker chose a very different context in which to develop his number theory and that this led him to fundamentally different views on the methods of number theory. 4.8 Kronecker—changing mathematical perceptions In discussing Dedekind’s work I have emphasized the manner in which he pursued mathematical ideas in consequence of his strongly held views on the desiderata he thought should govern mathematical argument. The view of Dedekind’s work thus obtained differs somewhat from that presented by other authors who begin from what may be described as Dedekind’s ‘philosophy of mathematics’ i.e. his views on the nature of mathematical objects and more particularly the notion of ‘set’ which he and Cantor were the first to promote. Similarly in discussing Kronecker’s work the emphasis here will also be somewhat askew from recent trends in interpretation offered by historians. Although many of Kronecker’s mathematical ideas have been championed by the great names in twentieth-century mathematics, it is only in recent years that historians have recognized the interest of his work. H.M.Edwards once again has played a leading role in this effort over the past decade in a series of detailed analyses of Kronecker’s ideas as presented in his ‘Grundzüge einer arithmetischen Theorie der algebraischen Grössen’ (‘Fundamentals of an Arithmetic Theory of Algebraic Quantities’)48 and related papers. Edwards focuses on the advantages that may be claimed for Kronecker’s approach to ‘ideal theory’ through his notion of ‘divisor’ as compared with Dedekind’s and relates this to Kronecker’s well-known ‘constructivist’ tendencies.49 From the point of view of the ‘working mathematician’, however, the picture of Kronecker as a precursor of constructivist or intuitionist mathematics that emerges can be difficult to reconcile with his well-known accomplishments in heavily ‘analytic’ areas such as the famous ‘limit theorems’, the so-called ‘Jugendtraum’ and other work on elliptic curves with complex multiplication, the distribution of numbers ‘modulo 1’ and so on.50 If, as is often said, Kronecker the ‘philosopher’ was in fact truly intent on reducing all mathematical constructions to computations with integers, how is it that Kronecker the ‘mathematician’ was so heavily involved in these (and other) areas

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where the principle of ‘continuity’ is ever present? Must this be viewed as merely an example of the working scientist’s pragmatism getting the better of his philosophical stance or can we find a way to see Kronecker’s mathematical thought as a unified whole? To avoid going over ground already ably covered by Edwards the Grundzüge will not be the centre of attention here. Instead I shall use material from his lectures on ‘General Number Theory’51 as an introduction to a discussion of the ‘Jugendtraum’ (and the so-called Kronecker-Weber Theorem) and some of the subsequent development of classfield theory (see the appendix to this chapter for a summary of the results and references). At the outset, it should be noted that mathematicians’ views of Kronecker have altered enormously in the past fifty years (views of Dedekind by contrast having remained quite stable). As late as 1950 André Weil found it necessary to begin a talk at the International Congress of Mathematicians52 on the notion of a Kroneckerian approach to number theory and algebraic geometry with a defence of his subject! Since that time the development of ‘arithmetic algebraic geometry’ has left no doubt as to the breadth and depth of Kronecker’s vision and insight. R.Langlands’ talk on ‘Contemporary Problems with Origins in the Jugendtraum’ at the Symposium on the ‘Hilbert Problems’ in 1974, tying Kronecker’s lifelong dream to Langlands’ own vast programme,53 served to clinch Kronecker’s new found position. An analysis of Kronecker’s views of mathematical argument may therefore also serve to illuminate the views of some of the contemporary mathematicians who find themselves walking in his footsteps. 4.9 Kronecker’s views on ‘number’ Kronecker regularly incorporated general discussions of the nature of ‘number’ into his annual course of university lectures and summarized these discussions in one or two published articles. In his ‘Uber den Zahlbegriff’ from 1887 he stated his views on the nature of number in a manner that could not be more at odds with Dedekind’s: I find that the most natural starting point for the development of the concept of number is found in the ordinal numbers. We use a supply of previously given and ordered markings with which we can distinguish various objects. The totality of the markings we have used we call the number of the objects and we attach the last marking we have used to this number
54

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Kronecker begins with the ordinal numbers (as opposed to Dedekind’s focus on cardinals) and with the symbols which do the counting, locating the concept of number in the way in which the symbols are employed. In contrast to Dedekind’s insistence on a ‘simple’ presentation of mathematical ideas which is independent of the ‘form of representation’ (to use Dedekind’s phrase) it is precisely the form of representation which is important to Kronecker. Behind this debate lies a broad disagreement as to the semantics of ‘form’ and ‘content’ which can be seen in many other contexts as well. In the same paper, Kronecker shows that the notion of ‘algebraic numbers’ as specific entities (in the case in point Kronecker is referring to real algebraic irrationals, for him a quite ‘abstract’ notion) can be avoided altogether. He replaces this concept with specific means of manipulating functions whose zero values demarcate intervals in the real line which have the property that they isolate an ‘algebraic number’ from its (Galois) conjugates. These techniques rely on elementary calculus (Sturm’s theorem and the intermediate value theorem) and algebra but do not introduce new concepts or types of number. He states that the possibilities of calculating with such irrationals can be reduced to the computations using these intervals and that therefore there is no need to introduce new entities such as ‘algebraic numbers’!55 In Kronecker’s view the ‘existence’ of such numbers is merely another way of saying that his calculations with intervals are possible. This example is typical of Kronecker’s views on number theory. Since he is of the view that numbers are merely systems of representations with which calculations are done, the essence of number theory is calculation. The means by which such calculation is performed may be (in modern terminology) strictly algebraic or may involve analytic tools from calculus or function theory. Kronecker is less concerned with the ‘purity’ of the techniques than he is with the ability to calculate. In another article he provides these reflections: Similar comments apply to the introduction of Dedekind’s concepts of Module and Ideal as well as to other ideas recently introduced (by Heine) to provide a completely general notion of ‘irrational’. In my view a general notion of, for example, an infinite series, can be allowed only where the terms or coefficients can be calculated following particular arithmetic rules and which can be tested in specific expressions
56

The focus on calculation and the forms or expressions which permit it thus also causes Kronecker to attack Dedekind’s requirement of generality. Kronecker is happy with methods from analysis so long as

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they are framed in terms of specific calculations and so long as they allow specific calculations to be undertaken. A general notion such as that of an ideal, which does not come with a specific means of computation, is not acceptable—not as a matter of taste, but because it is not mathematics as no calculation is possible with it. This approach to mathematical argument had a very striking influence on Kronecker’s work in number theory. 4.10 Kronecker’s number theory In his Grundzüge Kronecker lays out in a systematic way his theory of ‘Rationalitäts-Bereiche’ (‘rationality domains’– his response to Dedekind’s fields) which are the context in which he proposes to conduct his number theory. As Edwards has demonstrated in detail, these Bereiche or domains are constructed by an iterative process of ‘adjunction of indeterminates’ and taking quotients modulo various equivalence relations.57 Kronecker views this process as a vast generalization of the work of Gauss in studying congruences modulo a prime and he considers that the proper context for a generalized number theory (see below) would be the study, in general, of these domains.58 Edwards has shown that Kronecker’s theory of ‘Divisors’ in these domains generalizes Dedekind’s notion of ideals while maintaining certain advantages over Dedekind’s ‘non-constructive’ methods. In particular Edwards shows that Kronecker always views his domains as ‘relative’ to some base domain, i.e. as specific types of extensions, and that in this context the Kroneckerian focus on the generalization of the notion of ‘greatest common divisor’ (which is independent of the field or domain) is more appropriate than the Dedekindian notion of ‘prime ideal’ (which is not).59 It may be useful merely to note, at this juncture, that while Dedekind is clear that his ‘fields’ are ‘complete’ (with respect to the operations of arithmetic) and ‘whole’, Kronecker’s ‘domains’ are simply realms in which calculations of certain types are possible. At no time does Kronecker try to view his domains as ‘completed wholes’; his interest in introducing them is not to provide new types of numbers but rather to permit complex types of calculation to be undertaken in a systematic manner. The contrast in focus is made clear when we look at Kronecker’s approach to the problem in number theory which plagued Dedekind for two decades, the problem of ramification. Thanks to the work of Edwards and others we now have an edited version of a series of notes that Dedekind prepared but never published on Kronecker’s Grundzüge known as the Bunte Bemerkungen zu Kronecker’s Grundzüge (the ‘Miscellaneous Remarks’).60 These ‘Remarks’ give a wonderfully clear picture of the loci of disagreement

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between the two men of which they were both quite aware, but they also display a number of points of disagreement where it is clear that one party did not understand the ‘problem’ raised by the other side. One instance of this involves the following famous reference by Kummer in one of his great papers on cyclotomic numbers to work of Kronecker’s in 1858 which provided a generalization of the notion of ‘ideal number’ years before Dedekind published his ‘ideals’: I can [Kummer maintains] in this, as in connection with other general theorems which are common to all theories of ‘complex numbers’, refer to work of Kronecker which is shortly to appear, in which the theory of the most general types of ‘complex numbers’ in their connection to the splitting of forms of arbitrary degree is developed in its simplicity and generality.61 Dedekind was of course both sensitive and suspicious about this in view of the years of struggle the problems associated with ramification had cost him in coming up with what he viewed as a complete general theory. Yet Kronecker insists that he had such a theory in 1858— including an approach to the vexed problem of ‘inessential discriminant divisors’ (i.e. the existence of fields in which the index of every element is divisible by the same prime number)—but he held off from publishing precisely because he thought it incomplete in a wholly different sense, that it failed to deal with the problem of ‘associated genera’ or (as it has been known since Hilbert) class field theory.62 We can fictitiously reproduce Dedekind’s side of the story as follows. By theories of ‘complex numbers’, of course, Kummer is here referring to my number fields and the algebraic integers they contain. A complete and general theory of these fields requires my ideal theory in order to establish the equivalent of the property of unique factorization into primes which is the basis of the arithmetic theory of the rational integers. Since no simple rule for the behaviour of rational prime numbers when considered as elements of a general number field can be found, as the existence of inessential discriminant divisors shows, the theory of number fields must be developed simply on the basis of the properties of fields and integers without regard for the way in which the number fields may arise or may be ‘generated’. A complete and general theory will not concern itself with particular cases but rather provide statements which are true for all number fields, such as the finiteness of the class number or the structure of the group of units. At the time Kummer made his reference to Kronecker’s work Kummer could not have had this type of theory in mind since, by

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his own admission (as stated in the same paper in which the reference to Kronecker is made), he did not see how primes which divide the discriminant can be ‘truly called primes’. An equally fictitious outline of Kronecker’s side of the story might run as follows. The general facts to which Kummer makes reference, such as the finiteness of class number, are facts about equivalence classes of forms (as shown by Gauss and Dirichlet) and they can be generalized, as my Grundzüge shows, to forms in many variables and to forms with ‘coefficients’ which come from domains other than those of the rational integers. The domains do not exist in isolation but come about as a result of iterated processes of extension and reduction modulo equivalence relations. A general theory therefore requires that we understand how such extensions can take place and how to calculate in one domain if we know how to calculate in the domain from which it derives. We do not require statements which are true in every domain (in fact it is by no means clear what this phrase means) but rather how to move from one domain to another. Behind the semantic conflict (here about the meaning of the words ‘complete’ and ‘general’) mathematical history was being shaped. 4.11 A return to ramification Dedekind’s study of the discriminants of algebraic number fields in his great paper of 1882 is conducted in complete generality and his methods are as valid for the so-called relative case, i.e. the case of the extension of one number field by another, as they are for the absolute case, the case of a number field extending the rational numbers. This generality, as discussed above, is central to Dedekind’s thought. As striking as this characteristic is on first encounter with Dedekind’s work, there is another characteristic which is even more surprising to a contemporary mathematician, namely the absence of a number of lines of enquiry which today would be considered to be a natural part of Dedekind’s enterprise. These matters all have to do with specific types of extensions and their properties (it is typical, and in fact quite curious, that Dedekind makes no mention in this or his other major writings, of Gauss’s famous quadratic reciprocity!).63 The absence of these issues from Dedekind’s work almost ensures their central presence in Kronecker’s treatment of the subject precisely because they

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concern the study of factorization in certain specific types of extensions. We can best understand the nature of these lines of inquiry by considering three of the theorems with which Kronecker’s name is frequently associated. The first of these theorems states the basic fact that ramification is unavoidable for extensions of the rational numbers (some rational prime will always factor into the square or higher power of a prime ideal in the extension field), although there do exist unramified extensions of other ‘domains’ (Kronecker did not provide a proof, the first such was given by Minkowski in 1891).64 This theorem gives rise to two questions which guided much of Kronecker’s number theoretic work: • How can one construct the unramified extensions of number fields other than Q? • Can one find other extensions of Q, (or indeed other fields), which, though not unramified, behave as nicely as Kummer’s cyclotomic extensions in that one can easily predict the factorization of rational primes? The collection of extensions to which one naturally looks first to identify candidates exhibiting good behaviour as regards the factorization of primes are the so-called abelian extensions, i.e. those fields which are generated by the roots of an equation which has an abelian Galois group65 (the cyclotomic fields are of this sort). In fact, early in his life, Kronecker had found that all abelian extensions of the rational numbers are contained in extensions generated by the values of the circular functions (sine and cosine) at the values 2π(n/m), where n/m is some rational number. Put another way, any abelian extension of Q is contained in a cyclotomic extension (once again the proof was left to someone else, in this case Weber in 1886),66 thus providing at least a partial (negative) answer to the first question (the non-abelian case being still largely unknown). In addition to extending the work of his friend Kummer by showing the ubiquity of the cyclotomic fields, this discovery also set Kronecker in search of an answer to the second question, finding methods for constructing abelian extensions of number fields which are unramified so that the factorization behaviour of primes could be easily carried over from the smaller field to the extension. Today these extensions are known as class fields.67 Kronecker’s most famous discovery in this direction was the ‘Jugendtraum’ theorem which (in a simplified and technically incorrect statement) asserts that the unramified extensions of a quadratic imaginary number field (e.g. the Gaussian integers Z[i] described above) are generated by special values of certain elliptic

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functions (those with complex multiplication—see the appendix to this chapter).68 These discoveries (Kronecker himself did not supply correct proofs for any of the above) have much in common and in them we see all of the aspects of Kronecker’s thought in action. • First of all the domains in question are viewed as ‘relative’ to each other, Kronecker would represent each of them as an extension of a’base’ by adjoining a collection of indeterminates and then setting up equivalence relations. Ultimately the rational integers thus become the base for all domains and no ‘new entities’ are introduced in the passage from domain to domain, it is a purely algorithmic process. As noted above, Kronecker’s mathematical interest focuses not on the domains as completed entities but as realms or contexts in which calculations (such as prime factorization) can be studied. • The arithmetic properties of these new domains are also obtained by extension step by step. As noted above, the factorization of rational primes in cyclotomic fields can be derived from the factorization of polynomials modulo primes and the algebraic integers in these fields can be represented in terms of a ‘power basis’ made up of powers of the cyclotomic number which generates the field.69 The matter is somewhat more complicated when it comes to looking at extensions of number fields and leads ultimately to the notion of a class field as developed by Weber and Hilbert. The technical details here matter less than the fact that such class fields are extensions of a base field with (among other things) specified ‘good’ ramification behaviour so that factorization in the ‘top’ field is specified by that in the base field plus readily specified data about the extension. • Certain types of extension are studied, namely those referred to as ‘abelian’ meaning that the group of transformations of the larger field leaving the smaller one fixed (the ‘Galois group’ of the extension) is commutative. Since in Kronecker’s approach the elements of the larger fields are derived by adjunction of indeterminates and taking of congruences, these transformations take on a very concrete form as ‘groups of substitutions’ familiar from the algebraic theory of equations developed by Abel, Galois, Jordan and many others in the nineteenth century. Kronecker always considers particular types of extension. • In spite of their entirely algebraic provenance and definition these domains can be generated by certain specific values of certain (very special) analytic functions. Much of the power of these theorems derives from the care which Kronecker has taken to define his domains algebraically and constructively. This particular effort is rewarded when these algebraic objects are shown to be intimately

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related to special values of functions which have a purely analytic definition. The results would be diminished in equal measure if either non-constructive methods were allowed in constructing the domains in the first place or if the functions to which they are shown to be related were found to have a simple algebraic origin! Furthermore the consequences of such a relationship run, as it were, in both directions, so that in addition to providing a ‘transcendental’ means of generating number fields they also provide an ‘algebraic’ way to define ‘continuous’ quantities.70 4.12 Kronecker’s ‘general theory of arithmetic quantities’ As the above examples demonstrate, an exclusive focus on the constructive side of Kronecker’s thought, important though it is for an understanding of his views on mathematical argument, does not provide an adequate account of some of his greatest mathematical accomplishments. Throughout his work, side by side, the most ‘constructivist’ arguments are set over against results based on analytic function theory or similar techniques. In addition to the instances cited above, further examples are provided by André Weil’s intriguing musings on the possible reasons why, late in his life but freshly inspired by re-reading papers of Eisenstein which then had been neglected for fifty years, Kronecker adopted a new approach to defining elliptic functions in a series of memoirs (unfinished) for the Berlin Academy. Apart from the theorems on abelian extensions of cyclotomic and imaginary quadratic fields, Kronecker had already used special values of elliptic functions to describe solutions to Pell’s equation and to find solutions to the general fifth-order polynomial. His new techniques, based on very special types of infinite series introduced by Eisenstein, pointed the way towards even more spectacular results precisely because the series employed had a transparent algebraic structure and origin. As Weil shows, consideration of these series leads directly to results on the rationality of certain special values of so-called L-functions and hence to questions in arithmetic algebraic geometry and transcendence theory that are very much to the fore in current research programmes.71 On one reading of Kronecker it is surprising that he should have anything whatsoever to do with infinite series and related questions of convergence (i.e. the technical problems involved in making the series noted above ‘work’) just as it is surprising that he should have devoted so much attention during his career to elliptic functions. His wellknown fulminations about such things would seem to preclude a serious scientific interest in them. However, such a reading clearly does not do

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justice to Kronecker’s career as a ‘working mathematician’ and distorts his philosophical position as well. Kronecker is not a constructivist before time nor is he a forerunner of positions taken in the grand foundational debates of this century by the likes of Brouwer or Weyl. In fact Kronecker himself (rather late in his life) developed a framework in which to place much of his research, most of his university lectures and the better part of his writings, which he called his ‘theory of generalized arithmetic’. From the perspective of this broad framework it is possible to view the various parts of Kronecker’s mathematical career as a whole.72 Recall the discussion above in which Kronecker produces a method for rendering the introduction of real algebraic irrational numbers ‘unnecessary’. His procedure was to study such numbers, which are, by definition, the roots of algebraic equations, by proving that one can approximate the roots of the equation by intervals of real numbers which can then be manipulated. The process of approximation introduces no ‘new’ entities and relies solely on algebraic facts and Sturm’s theorem (and is more than superficially similar to the notion of ‘o minimality’ which is currently under active investigation by model theorists).73 Kronecker wishes this to be thought of as a generalized form of arithmetic in which algebraic means are employed to analyse phenomena which are apparently of a continuous nature. In certain of his papers and in his university lectures he goes further and discusses the possibility of using congruences themselves to create approximations by considering solutions to congruences modulo higher and higher powers of the modulus. In this manner a notion of ‘quantity’ or ‘valuation’ can be derived which has a totally algebraic origin. This is now known as ‘p-adic’ analysis. While the idea is already evident in Kummer’s later work and Kronecker’s student Hensel used it to great effect to provide a new proof of Dedekind’s discriminant theorem in 1894,74 Kronecker himself published relatively little using these techniques. The little he does say, however, indicates that he was fully aware of the possibilities and should be taken together with his insistence on rendering irrationals ‘unnecessary’ by similar techniques in the ‘archimedean’ case. This purely algebraic process provides a notion of ‘size’ or ‘quantity’ which otherwise might seem bound up with notions of continuity that are not algebraic in origin. The process can be carried out not only with numbers with respect to a prime but also with forms in one or more indeterminates and with respect to another form (this is now known as ‘formal completion’). From this perspective the quantitative notions traditionally applied to ‘real numbers’ have no pride of place. These too should be seen as arising from an essentially algebraic process of ‘completion’. The generalized arithmetic then becomes remarkably

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similar to current notions of ‘local places’ of ‘global fields’ and justifies the support that Kroneckerian ideas have had in this century from the likes of Weyl and Weil. Using the full range of modern techniques, algebraic geometers in the twentieth century led by Weil and Grothendieck have constructed a’Kroneckerian’ theory of ‘generalized arithmetic’ in which full freedom is granted to the mathematician to create geometric objects using any ‘coefficients’ desired and to consider an enormous range of possible ‘valuations’ and ‘topologies’ on these objects. To paraphrase Weil, all of this was clearly far beyond Kronecker’s ability, but perhaps not beyond his insight. In his ‘generalized arithmetic’ the continous aspect of a ‘quantity’ was not divorced from its discrete aspect as an algebraic expression to be manipulated. The introduction of new entities, such as Dedekind’s ideals or algebraic numbers, was rendered unnecessary by the use of new techniques of calculation and manipulation. Expanded powers of calculation not only ‘justified’ in a philosophical way the use of these new entities but, more importantly, linked them back to the algebraic domains providing new methods for generating and computing them. This version of the ‘arithmetization’ of mathematics broadens rather than narrows the range of concepts it seeks to underpin.75 We therefore can see Kronecker’s use of analytic techniques in number theory not as examples of scientific pragmatism triumphing over philosophical rectitude but as the fulfillment of deeply held and well-articulated views on the nature of mathematical argument. In theorems such as the ‘Jugendtraum’, algebraic methods provide contexts in which ‘continuous quantities’, such as the values taken by elliptic functions, can be placed and computed algebraically and can be analysed arithmetically. Kronecker’s overriding concern at all stages is in the manner in which numbers or ‘quantities’ are represented and hence the computations in which they can be involved. His ‘domains’ are the broadest possible contexts for such computations and they include data of both ‘continuous’ and ‘discrete’ sorts. At the end of his paper on ‘The Concept of Number’ Kronecker states: hence the results of the deepest mathematical researches must be representable in terms of the properties of the integers
. The fruit of these efforts, our number words and symbols, is not only the precondition for today’s number theoretic investigations or for the ‘laws by which our knowledge is able to govern the movements of the stars’ but it is also the preconditions for the forms that practical life takes today, the unprecedented spread of commerce and travel which so profoundly differentiates our world from that of our ancestors.76

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The famous Kroneckerian emphasis on relating mathematics back to the integers is of course clearly visible in this passage, but it should now be apparent that this emphasis does not preclude consideration of the realm of the ‘continuous’ but forms the foundation for a broad view of the nature of ‘arithmetic’ in which algebraic operations are used to generalize notions of quantity, a view which has not ceased to inspire succeeding generations of mathematicians. 4.13 Conclusion For Dedekind only a general method, capable of considering all entities of a specific type at one time, is adequate to grasp the mathematical idea which underlies a given subject matter. Simplicity of formulation providing maximum transparency of this idea is required. For Kronecker a mathematical idea is a computation and computation is adequate to all mathematical ideas. Generality is provided by establishing domains in which the broadest range of computation with the broadest range of formulations is provided for. Dedekind’s focus on simple ideas and a dislike for ‘mere calculation’ was taken up by Hilbert and influenced all of his work from number theory to foundations of mathematics. It also led Dedekind (together with his friend Cantor) to the idea of ‘set’ and more generally to the notion that there were certain simple and basic ideas which lay at the foundation of mathematical thought. The later career of these ideas in the hands of Hilbert, Bourbaki and others is only too well known. In all of this activity Kronecker’s ideas came to seem out of date. Attitudes began to change in the 1920s. Weil’s frustration with the inadequacy of Dedekind’s ideals for purposes of number theory was expressed in print as early as 1928, but the development of an adequate framework for arithmetic algebraic geometry took another 30 years. Logicians too, somewhat disillusioned by the failure of the foundationalist programme promulgated by Hilbert and stimulated by more recent interest in ‘computation’, have revised previous opinions of Kronecker’s position. In considering the work of these two men it is important, however, to go beyond a narrow focus on foundationalist or logical concerns and attempt to understand their mathematics as a whole. Neither man would have recognized himself or his thought in the stereotyped positions that came to be taken in the great debates on the foundations of mathematics of 1900–1930. Indeed Hilbert’s views incorporate as much of Kronecker as they do of Dedekind (not to mention Cantor) and to assimilate his views to any one of his forebears or vice versa is to distort the thought of one or both.

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Given the philosophical antagonism of the two men it is important to try to find a point of view which allows their work to be compared without hidden prejudice or preconception. The notion of ‘context’ has been the starting point of the discussion presented here. Dedekind’s fields and Kronecker’s bereiche are contexts of very different kinds adopted for the common purpose of extending arithmetic and algebraic notions beyond the integers and rational numbers in which they originated. To conclude this portion of the investigation we compare the use that these mathematicians made of their contexts to Euclid’s contexts. Dedekind defines fields in which new entities are related by means of the known operations of arithmetic and the elements in the fields are either integral or fractional. Kronecker defines bereiche in which previously given symbols are manipulated in new ways and an algebraic meaning is given to the notion of quantity. Euclid introduces neither new entities nor new operations but rather new means for comparing and relating known things in ratios and proportions. Paradoxically Dedekind’s ‘non-finitistic’ methods end up with the notion of ‘measure’ or quantity absent from his number fields in which all elements are either integral or fractional (as in Euclid’s Book VII) while Kronecker’s finitistic methods aim at reintroducing notions of quantity through algebraic means (as in Euclid’s Book V). Euclid, however, does not distinguish between the discrete and the continuous or between numbers as quantities and as algebraic symbols. Indeed he does not introduce notions such as integer or rational number at all. The novelty brought into his argument by introducing the context of measurement does not lie in the use of new entities or operations but in the increased range and precision of the statements that can be made about already known and defined things. As a result his discussion incorporates distinctions between part as a means of measurement and part as a means of construction as well as the ability to relate these two functions in the notion of incommensurable. Dedekind seeks generality as a means of defining and grasping a single mathematical idea in its many manifestations. His theory is general in that it addresses all number fields ‘at once’ and does not distinguish between them based on their mode of generation. Kronecker seeks generality to demonstrate the true nature and scope of computation and to obviate the need for the introduction of new entities. His theory is general in that it allows for computation in the broadest possible range of circumstances. Euclid seeks generality to be able to make a wider set of comparisons between already defined things and therefore to provide the ability to ‘say more’ about them and to increase the precision of his science without thereby committing himself to the existence of new objects or requiring new constructions.

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Both Dedekind and Kronecker were caught up in the nineteenth century notion that mathematics could be made more rigorous by ‘arithmetization’, i.e. by minimizing or even eliminating the use of (what came to be known as) ‘geometric intuition’. Dedekind saw mathematics ultimately as part of logic, the laws of thought, and therefore to be made rigorous by providing simple definitions and building mathematical arguments from them. Simple ideas, stated through definitions, general enough to cover all instances of a phenomenon and manipulated axiomatically were the key to success in his view. Kronecker saw mathematics as a science on its own, separate from logic and based on calculation. Clearly established symbolic systems, general enough to encompass all manipulations required were his requirements for extending the familiar computations of elementary number theory and exploring the new realms thus opened up. In contrast, Euclid’s mathematics progresses by increasing the precision of the statements it can make about its subject matter. New techniques are introduced not to ‘generalize’ existing methods but to allow the mathematician a higher level of ‘discernment’ in connection with the objects of his study. The result is not a mathematics in which the discrete and the continuous, the rational and the irrational, the arithmetic and the geometric are opposed to one another so that one is used to explain or exclude the other, but one in which the range of comparisons available to the mathematician is increased to the maximum extent possible. Appendix This appendix is included to provide an easy reference for some of the basic concepts in algebraic number theory encountered in the review of Dedekind and Kronecker’s work. There is no pretention to completeness of any kind and the reader who requires further details is urged to consult one of the many standard texts on number theory. We define a number field K to be a field extension of the rationals Q of finite degree, written [K:Q] < , where ‘degree’ means simply the dimension of K as a vector space over Q. This definition is essentially that of the fourth edition of the eleventh supplement to Dirichlet— Dedekind in which Dedekind has abandoned earlier versions where the algebra structure of K over Q is highlighted instead of its structure as a vector space. In the fourth edition Dedekind discusses linear transformations of K which leave Q fixed and shows that there is always which are a collection (in fact a group) of n such transformations independent in the sense that no equation of the form

110 NUMBER THEORY IN THE NINETEENTH CENTURY

where the (and non-zero) is possible (such an equation being understood to say that the sum on the left-hand side is the zero transformation when applied to any element of K). For an element K we on k. The simplest way to construct these write k(i) for the effect transformations is to note that any such K is generated by an element γ, i.e. K consists of numbers of the form where d is the degree of the extension and γ satisfies an algebraic equation of degree d with coefficients in Q. Permuting the roots of the equation for γ then gives the desired transformations (this approach, however, deviates from Dedekind’s mature views which would see in it ‘an accidental form of representation’ rather than an analysis based on the simplest form of an idea). , is the set of elements The ring of integers of the number field, in K which satisfy a monic equation with rational coefficients: The proof that these elements form a ring is due to Dedekind in general (although special cases had been shown by Eisenstein and others). As stated, these definitions are ‘absolute’ (over Q) but can easily be modified to give the ‘relative’ case of L over K, one number field extending another. An ideal / in this ring is a collection of elements which satisfy the conditions: and As Dedekind pointed out, an ideal is a special case of a module which is a collection of elements of K, M, which satisfy the same conditions: and A module M is called a ‘fractional’ ideal if there is a such that . An ideal is prime if . As discussed in the text, Dedekind showed that any ideal in R can be ‘factored’ in an essentially unique way as the product of powers of prime ideals in much the same way as an integer can be factored into prime numbers. Kummer had originally considered the rings Z[ζ] obtained by adjoining ζ an nth root of 1, to Z, i.e. numbers of the form a+bζ. He did not consider the number fields obtained by replacing a and b by rational numbers for the rings of integers and Q(ζ) for the number field). Had he done so he would not have arrived at any new information because, as noted in the text, the elements in are precisely Z[ζ]. This is not so

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in general. For example, if we consider the where d is some , where then the integer (i.e. numbers of the form will be (i.e. numbers of the form ring of integers only if (mod 4) and not if (mod 4). This is where Dedekind’s new context starts to produce new phenomena. Kummer’s cyclotomic numbers do illustrate another aspect of number fields however. Note that the definition of an ideal is patterned after the sequence of rational integers divisible by a given integer (e.g. the multiples of 3). In some instances it is the case that all ideals are in fact multiples of some integer (perhaps not a rational integer but none the less an integer from the ring of integers of the field). Such rings are called principal ideal rings (and ideals which are multiples of a single element are called principal ideals). Kummer showed by calculation that Z[ζ], where ζ23=1, was not a principal ideal ring! By the third edition of Dirichlet—Dedekind, Dedekind had developed a notion of multiplication of (fractional) ideals and he was able to show that the ideals of the ring of integers of any number field form a group (i.e. the product of two ideals is an ideal and for every ideal I there is an ideal I −1 such that I(I-1)=e where e is the identity ideal—e whole ring ). If we consider two ideals I, J to be equivalent if there is a principal ideal P such that IP=J, then the collection of (an ideal of the form (a) for ideals falls into a collection of equivalence classes. Using Dirichlet’s reworking of Gauss, Dedekind was able to show that the number of such ‘ideal classes’ is finite for the ring of integers of any number field.77 A ring is a principal ideal ring if and only if the number of these classes is 1. We see therefore that in general an ideal is not the collection of multiples of a single number but may be represented as a collection of numbers of the form for and Similarly a module (technically a Z module) can be represented as a sum of products this time with the xi in K. Such a collection of xi is called a set of generators for the ideal or the module.78 Consider for example the ring of integers of the field K. This is a module with a set of generators {ai}, i=1, 2 3,
, n. If we consider the conjugates of these generators we get an array of elements αi(j) which form a matrix. The determinant of this matrix is called the discriminant of the field (Dedekind called it the ‘grundzahl’) written DK. We can do the same for any ideal or module. In particular if we consider the ideal formed of multiples of an integer m, we get a discriminant D(m). This number is related to DK by an equation D(m)=(DK)(index m) where index m is the index referred to in the text (it is an integer, in fact the square of an integer, and the above equation can be taken as its definition).

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As discussed in the text, Dedekind was able to show that the factorization of an ideal generated by a rational prime p, when considered as an element of the rings of integers of a number field K, followed the pattern discovered by Kummer if the number field K is generated by an element in its ring of integers, say m, and p does not divide the index of m. Of course, you can hope that by astutely changing m into another integer, say m’, which also generates K you are never in the position where the theorem fails. Dedekind tells us that this was where θ satisfies: precisely his hope until he found the field For any index m is divisible by 2.79 It was from this point on that his theory began to take on its final form. We can now give a simple definition of ramification.80 A prime number p is said to ramify in a number field K if the ideal (p) of multiples of p in the ring of integers K factors into a product of prime ideals where at least one prime ideal appears to a power higher than 1. Dedekind, in his great paper of 1882, was able to • show that p ramifies in K if and only if p divides the discriminant of K, DK, and • determine the exact powers of the p in the factorization of DK assuming that p did not divide the power to which any of the prime ideals in its factorization appeared, but he was unable to give any conclusion if p did divide one of these powers. It was the inability to complete this determination which most likely kept him from including the full ramification theory in any of the editions of Dirichlet—Dedekind. There is a version for the relative case of two number fields as well as the absolute case discussed here. Returning to Kummer’s cyclotomic integers, they exhibit another, quite different property as well. The linear transformations {øi} noted above all take Q(ζ) to itself. In this case we say that the extension Q(ζ) over Q is ‘Galois’. The {øi} form a group which is known as the Galois group of the extension. In the case of the cyclotomic numbers this group is abelian, i.e. the product of two elements is the same regardless of which is taken first. One of Kronecker’s first major discoveries was that any number field K over Q which was Galois and abelian was actually contained in a cyclotomic field! This was the beginning of Kronecker’s efforts to describe extensions of number fields based on data such as the nature of their Galois groups or the absence of ramification. The problem of describing the possible extensions of Q or of an arbitrary number field which satisfy certain restrictions on Galois groups (abelian) and ramification (a very sophisticated system for prescribing allowed ramification was developed by Weber, Hilbert and others)

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became known as class field theory. By comparison, very little is known about possible extensions with non-abelian Galois groups. The cyclotomic extensions of Q are generated by the roots of unity which are in turn given by the values which a special complex analytic function takes when evaluated at certain points. The function in and the points are the rational numbers 1/n (since question is The Kronecker—Weber theorem thus gives an arithmetic meaning to this analytic function, or at least to its values at certain special points. This is a very beautiful and surprising fact. Kronecker’s great dream, which has vastly increased in size with the advent of Langlands and the development of Iwasawa theory, is simply that this is not an accident. In other words, Kronecker thought that number fields of special kinds (or, as we would say today, class fields or maximal unramified extensions) could provide an arithmetic meaning for the values at special points of certain geometric objects or certain special analytic functions. The case which he considered most fully is that of the unramified abelian extensions of quadratic imaginary fields. These are fields L, which are abelian extensions of a field K, which is generated over Q by √( − d)where d is a positive integer where no prime ideal in K appears in the relative discriminant of L over K. In the ring of integers of K (call it O) there are sub-rings, called orders, which are those sub-rings generated by (1, nα) where (1, α) generates O as a module over Z and n is an integer. The ideal classes of these orders (i.e. the group of equivalence classes of ideals where two ideals are considered equivalent if they differ by a principal ideal) correspond in a one-to-one fashion with certain algebraic curves (specifically, elliptic curves with complex multiplication). Furthermore these curves are classifed by the values of a certain complex analytic function, known as the j function. If we take n=1 above then Kronecker’s Jugendtraum theorem states that the value of j associated to an elliptic curve corresponding to an element of the ideal class group of K generates L (there are more complicated versions where n  1). Kronecker never published a full proof of this wonderful theorem. Hilbert, in the twelfth of the problems that he set before the International Congress of Mathematicians in 1900, asked how far this principle of generating special number fields by the values of complex analytic functions could be extended. Few today would hazard a guess.

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Part III Mathematical wholes and the establishment of generality in Euclid, Weil and Grothendieck

116

INTRODUCTION In Parts I and II, I have shown in some detail that the term ‘part’ plays an extraordinary and multifaceted role in Euclid’s argument throughout Books I–IV and V–X of the Elements. Initially, ‘parts’ and their presence or absence are the key to the notion of definition and what is definable, providing a link between the discursive arguments of mathematical demonstrations and the diagrams in which they are presented. In Books V–X the term ‘parts’ provides a (series of) context(s) in which things which are defined and known by other means can be compared with one another in a uniform manner. Thus in each of the first two stages of the argument of the Elements a specific role is given to the term ‘part’ and its derivatives. In Books XI–XIII the term ‘part’ can be seen in its final and perhaps most ‘natural’ guise, in the pair ‘part and whole’. The reader may have noted that although hitherto there has been much talk of parts there has been relatively little talk of parts as parts of something. The introduction of solid figures in Book XI addresses this concern. Books XI–XIII provide a completion to the argument of the Elements in a variety of senses. The solid figures which are their subject provide wholes which are made up by the figures of Books I–IV and these complex wholes and their parts can be related to one another by means of the techniques of comparison in Books V–X. There are no ‘further’ or ‘higher dimensional’ figures beyond those presented here because no regular types of solid figures can be singled out (in Euclid’s approach) to correspond to straight lines and plane surfaces. Instead a new type of ‘regularity’ is introduced, based on the parts which make up the figures, the so-called regular solids, and the types of such figures are shown to be limited in number. Solid figures provide a completion to the study of figures in general both because they serve as wholes for their parts and because they fall into a limited number of types (compared, for example, with the unlimited sequence of regular polygons glimpsed in Book IV). At the close of the discussion of Book XIII it will be clear that Euclid has provided a completion of his investigation of the subject matter outlined in the Definitions in Book I. The notion of completeness has become of great importance in contemporary mathematics as mathematicians have sought to establish theories which generalize a wide variety of previously studied phenomena and which seek to provide a unified treatment of a range of seemingly disparate subject matters. This type of enterprise has been of particular importance (and perhaps even notoriety) in the area of algebraic geometry especially as it relates to number theory. It is natural, therefore, to consider the work of André Weil and Alexander Grothendieck, two of the most important mathematicians of the twentieth century, as they have sought in different ways to broaden the

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algebraic geometric frameworks available to mathematicians with a view to encompassing arithmetic as well as geometric applications. Naturally there is no intention here to provide a full exposition or explanation of the work of these two mathematicians. Apart from the usual excuses of lack of space it must also be said that mathematicians today are far from a full understanding of the implications of this work and that a complete and balanced assessment must await future investigation.1 The discussion is limited to a few points of textual and argumentative strategy which highlight notions of completeness, generality and the concept of ‘wholes’ in mathematics. In addition to its functioning in the argument presented here, I hope that this brief sampling of the ideas of these remarkable men will serve to arouse curiosity about this work in a wider readership. The lack of broad discussion about contemporary mathematics is evinced by the fact that, although some of the work discussed below dates as far back as the 1920s, and although it is of undoubted importance and influence, it has not received much attention outside of purely mathematical circles. As a final introductory point, the reader must be prepared for a more mathematically technical presentation than has been given up to this point. Particularly in Chapter 6, where certain issues of interest in current research are discussed, it has not been possible to provide a thoroughgoing explanation of all of the terms and concepts employed. The central thrust of the argument, however, does not involve complete immersion in the technical details and those who find the discussion mystifying in places will, it is hoped, persevere to the conclusion. Reversing the order followed in previous chapters I shall begin with a discussion of the work of the two contemporary mathematicians prior to concluding the analysis of the Elements.

5 Types of wholes

5.1 André Weil and arithmetic algebraic geometry In his thesis (published in 1928) Weil reproved and greatly extended a result of L.J.Mordell concerning the group of rational points on an elliptic curve. Known today as the Mordell-Weil theorem it remains a cornerstone of the subject now known as Diophantine Geometry and it is not too much to suggest that Weil was the first to recognize the existence and importance of this new subject matter.1 It had been known since Poincaré (1901) that the rational points on an elliptic curve form a group under the group law of the curve itself. Mordell (1922) showed that this group was finitely generated2 when the equation defining the curve was taken as having rational coefficients. Weil’s result extended Mordell’s in two directions: on the one hand he showed that the result remained true if the field of definition of the curve was extended from the rational numbers to any number field (one then notes that the points on the curve with co-ordinates in this field once again form a group and Weil shows this group to be finitely generated) and on the other hand he showed that this extended version holds not only for elliptic curves but also for their higher dimensional analogues, abelian varieties. Mordell’s work had been based on an ingenious use of the technique of ‘descent’, which can be traced back (at least) as far as Fermat, using division (bi-section in fact, an example of ‘2-descent’) of elliptic functions. Working with elliptic curves over the rational numbers one has very explicit equations to handle as well as a wide range of formulae for manipulating elliptic functions. This is no longer the case when one considers abelian varieties. Weil was therefore faced with a series of difficulties to overcome in generalizing Mordell’s work and both in his approach to the problem and in the specifics of his solution we can see the genesis of the idea of an ‘arithmetic algebraic geometry’.

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In the introduction to his thesis3 Weil suggests that while (projective) geometry deals with properties which are invariant under rational transformations,4 regardless of the field over which these are defined, his new arithmetic geometry will deal with properties which are invariant under rational transformations defined with coefficients coming from the rational numbers (or a specified finite extension of the rational numbers, i.e. a number field). Hilbert and Hurwitz as well as Poincaré had also seen the possibility of this kind of approach and had begun with curves—Hilbert and Hurwitz for genus 0 and Poincaré for genus 1—and specifically elliptic curves (as Weil notes, by regarding the problem as a part of geometry one is led naturally to consider the genus of a curve rather than the degree of an equation as the basic parameter).5 Since each curve of genus 1 or higher is associated with a specific abelian variety, its Jacobian, Weil views his results as extending this previous work to curves of any genus, thus providing a general result in the arithmetic algebraic geometry of curves. To accomplish this objective Weil requires new tools, notably his ‘théorème de decomposition’ or, as one would say today, the theory of ‘heights’, which uses geometric and algebraic means to estimate the arithmetic ‘complexity’ of a point on a geometric variety.6 In place of the highly explicit elliptic functions which Mordell has at his disposal, Weil employs the analytic theory of theta functions and their division (bi-section once again) but notes that a purely algebraic approach to this side of the problem should be possible as well. In a footnote he remarks that the relatively underdeveloped state of algebraic techniques in this area at that time is not surprising, ‘in view of the fact that transcendental methods are so well adapted to the study of abelian functions [so that] no one has ever tried to study them using purely algebraic procedures’. In his attempts to study the arithmetic properties of these functions, Weil was breaking new ground.7 Within a year of the publication of Weil’s thesis, C.L.Siegel had used some of the results to prove his famous theorem that an affine curve of genus 2 or greater has only a finite number of integer points.8 While there is no doubting the strides that have been made since Weil’s thesis and Siegel’s subsequent efforts, the extent to which this initial work defined the area now known as Diophantine Geometry can be seen in the fact that many of the most important and natural questions, given below, which were raised explicitly or implicitly in the work of Mordell, Weil and Siegel remain unanswered: • With Mordell’s result in hand, the next natural question is that of the rank (i.e. the number of generators) of elliptic curves or, more generally, abelian varieties over the rational numbers. This is the subject of precise conjectures9(which have been shown to be true in

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certain special cases as well as having been tested by extensive computer calculations) but a full proof remains out of reach. • Following Weil’s generalizations one can ask for the rank of abelian varities over number fields. This question has barely been touched. • Following Siegel’s result one can ask for a procedure (i.e. an algorithm) to determine whether a given affine curve has any integer points (or, more generally, whether a given projective curve has any rational points). Although Faltings has greatly advanced matters by proving that projective curves of genus 2 or higher have at most a finite number of rational points, this most fundamental of questions remains open, apart from Baker’s results in genus 1 and variants which apply to certain other (specific) types of curves. In certain respects this ‘slow’ progress must be attributed to the ‘early’ appearance of these important results. It took decades for a systematic approach to the mathematical subject matter that Weil (and Siegel) outlined to be discovered. The next steps were again due to Weil. 5.2 Weil and the Riemann hypothesis One of the outstanding legacies of Riemann’s work on functions of a complex variable was the setting of the theory of the ‘zeta function’

on a firm foundation. Riemann showed the importance of the study of this function for a range of problems in number theory centering around the distribution of prime numbers, and he further demonstrated that many of these problems could be settled if one knew the location of the zeros of this function. In spite of continued assaults and much progress since Riemann’s initial investigations this tantalizing question remains one of the major unsolved problems in mathematics. To note its relevance to Weil’s work we must take a small detour. In his thesis (published in 1920) the German mathematician E. Artin10 had developed the arithmetic theory of ‘function fields over a finite field’, in particular the field of functions on a curve over a finite field, and noted the many similarities with the theory of number fields developed by Dirichlet, Dedekind, Kronecker and Hilbert. The analogies between function fields and number fields had been known since Dedekind’s time (at least in characteristic zero), but Artin’s work was perhaps the first to take the base field to have positive characteristic11 as opposed to subfields of the complex numbers. This required an entirely algebraic development of the subject since the

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transcendental techniques derived from working over the field of complex numbers are inapplicable in this new context. Artin also (later) developed a quite general theory of L-functions which, once again by purely algebraic means, defined functions akin to the zeta function for general number fields and for function fields.12 Artin may thus be seen to have been working to ‘geometrize (or at least “algebraicize”) number theory’ while Weil was trying to ‘arithmetize geometry’, and Weil has remarked on the excitement with which he and his colleagues in those days awaited new numbers of the journals which regularly contained Artin’s work. These results showed that a ‘Riemann hypothesis’ could be formulated for the L-functions arising in function fields. In an astonishing piece of work (announced in 1940 but only fully written up in 1948) Weil was able to use geometric techniques to prove the ‘Riemann hypothesis’ for the function field of any curve13 over a finite field. In the course of this effort he convinced himself that a reformulation of the foundations of algebraic geometry was imperative.14 His approach to the ‘Riemann hypothesis for finite fields’ was to use the theory of ‘correspondences’ on an algebraic variety which had been developed by Severi and the ‘Italian school’ of algebraic geometers.15 Correspondences on a curve give rise to transformations of the associated abelian variety (the Jacobian of the curve)16 into itself. Weil showed that these could be described by matrices with l-adic entries (l being a prime number different from the characteristic of the base field) and that simple algebraic properties of these matrices were sufficient to prove the results. It is only fitting that, years later, when systematic theories generalizing Weil’s approach were developed, they were entitled ‘Weil cohomology theories’. But Weil was unhappy with the state of algebraic geometry as he found it. While there was no question as to the validity of the results on which he based his proof, the methods of the subject seemed to ignore problems which arose in many of the cases he needed to consider in examining correspondences or, more generally, in studying ‘intersection theory’ on an algebraic variety. Weil was not alone in finding fault with algebraic geometry as it had been practised up to that time and a number of other mathematicians, most notably perhaps O. Zariski in the USA, were undertaking similar projects at the same time. The outcome of this work was a monograph, Foundations of Algebraic Geometry, written in English and published by the American Mathematical Society in 1946. It is a forbidding tome and since its language has now been largely superseded (as will be discussed below) it is not widely studied. Nonetheless it contains much of Weil’s most systematic thinking about geometry and merits close consideration. The following discussion merely skims the surface of this text.

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5.3 Weil and The Foundations of Algebraic Geometry In their pioneering studies of algebraic varieties and, in particular, the behaviour of intersecting subvarieties, the distinguished Italian algebraic geometers of the late nineteenth and early twentieth century had been led to distinguish between cases of ‘general position’ and ‘degenerate cases’. The thrust of this distinction is most easily seen if one considers families of varieties which vary (or as one often says ‘move’) according to a parameter (or set of parameters). Consider the family of curves y2 = x3 + tx−t, where x and y take complex values and t is considered as a (real) parameter. For values of t ≥ 0 this produces a nice collection of smooth curves varying smoothly. When t = 0, however, we have the curve y2 =x3 with a ‘singular’ point at y = x= 0 (taking derivatives and denoting ` /` x by a prime we have 2y (y’) = 3 x2 (x’) + t so that y’ and x’ cannot both be zero if t  0; both can, however be zero if t= 0). If we now consider the intersection of these curves with a line passing through (0,0), say x = y, we see that as t → 0 distinct points of intersection on the ‘general’ (or ‘generic’) curve where t  0 coalesce to form a singular point17 on the ‘special’ curve y2 = x3 where t = 0 (see Figure 5.1). In the older language, the intersections on the curves with t  0 are in ‘general position’ whereas the intersection at t = 0 is ‘degenerate’. The reader is invited to imagine the more complex situations that can arise with equations of higher degree and with varieties of higher dimension (more variables). An algebraic geometer considers the intersection of the line x=0 at the point (0, 0) on the curve y2 = x3 to be somewhat ‘more than’ a point in the following sense. Viewed simply as a set of points, the curve includes (0, 0) along with all of its other points. The above remarks suggest, however, that there is some additional structure which singles out (0, 0) as special in that it is in some sense the combination of two points. From Zariski, Weil had available to him a rigorous definition of the notion of ‘simple point’ using the algebraic concept of ‘integral closure’. Further developments in algebra since Zariski’s time have enabled the development of a full-scale ‘local intersection theory’ which describes the geometric phenomena illustrated above in purely algebraic terms. Weil states clearly, however, that he does not think that algebraic geometers should risk losing themselves in what he terms the algebraists’ ‘makeshift constructions full of rings, ideals and valuations’, but should strive to return as quickly as possible to ‘the palaces which are ours by birthright, to consolidate shaky foundations, to provide roofs where they are missing, to finish, in harmony with the portions already existing, what has been left undone’.18 How then does

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Figure 5.1 Formation of a ‘double’ point

he propose to deal with the many problems of ‘multiplicity’ whose difficulty the simple example given above only hints at? In outline Weil’s solution is to distinguish between the basic object itself—the variety—and collections of subvarieties which may be singled out, so-called ‘chains’ and ‘cycles’ on the variety. The basic object is thus kept close to the intuition of the older geometers and does not require elaborate algebraic tools to define. The complex collections of subvarieties are studied separately and are not defined under the same heading. Weil argues that this approach can provide the full generality required to deal with arithmetic algebraic geometry. The first step in the argument is the introduction of a ‘universal domain’ in addition to the base field over which the objects to be studied are defined. Whereas previously geometers had defined their objects over various fields Weil requires that, in addition to the base field of definition, a large ‘universal domain’ be given which encompasses all of the fields which may show up in any of the constructions to be made over the base field. Hence the universal domain must have ‘infinite transcendence degree’ so that it can encompass varieties of arbitrarily high dimension over the base field. It is also ‘algebraically closed’ so that the fields generated by the

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co-ordinates of any point (or set of points) of any variety will also be contained in it. The base field and the universal domain may have characteristic 0 or positive characteristic.19 While the use to which Weil puts these domains will be discussed shortly, a number of limitations of this approach may be noted. One type of problem arises in arithmetic questions when one wishes to ‘reduce’ a variety modulo a prime. As Weil himself points out in his comments to the second edition (1962) of his book,20 in such questions it may be necessary to use two universal domains to take account of the phenomenon of reduction when the reduction involves the introduction of objects over fields of differing characteristic.21 A further question arises as to the independence of the geometric theorems of the choice of universal domain. Weil labels this a ‘metamathematical’ question and argues that the theorems of algebraic geometry will not vary with the choice of universal domain so long as it has infinite transcendence degree and is algebraically closed. He justifies his assertion (at least as far as geometry in characteristic zero is concerned) by reference to the so-called Lefschetz principle. Attempts to make this argument rigorous will rely on the model theoretic notion of ‘model completeness’ as developed by Abraham Robinson and his mathematical progeny and will further depend on all of the statements involved being in the nature of ‘first-order’ propositions from the point of view of logic.22 These questions become more difficult in positive characteristic so that it cannot be said that a fully satisfactory response has been given to this question at present. Putting aside these objections, we see that the thrust of Weil’s argument is that all phenomena of algebraic geometric interest can be studied within the confines of a chosen universal domain and that the choice of one or the other of these does not (should not) influence the mathematical outcome. On the other hand by fixing a base field and a universal domain it is possible to specify an ‘absolute’ notion of ‘irreducibility’ which is the key to his definition of variety. If we return to the example given above we can see that the intersection of the line x = y with curves y2 = x3+tx−t for comes in two separate ‘pieces’ (i.e. consists of two separate points) and so is not ‘irreducible’. This is carried over to the case when t=0 so that even though it would appear that the intersection consists of only one point we must regard it as in fact being made up of two points which happen to be in the ‘same place’ (and we say that it is ‘unreduced’). Weil reserves the term ‘variety’ for reduced and irreducible objects so none of these intersections can be considered to be a variety in itself or even a subvariety of the original curve. He then uses the term ‘chain’ instead to describe arbitrary collections of subvarieties and ‘cycle’ to describe arbitrary collections of non-singular

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Figure 5.2 Reducible and irreducible curves

subvarieties. In the example we are using (Figure 5.1), we see that on each curve in the family we have marked out a pair of simple points. Since a simple point is irreducible it is a subvariety of the curve on which it sits, so, in other words, we have marked out on each curve a cycle consisting of two points. For the curve y2=x3 we have a chain23 consisting of a single point counted twice (i.e. with ‘multiplicity’ two). Now the notion of ‘irreducibility’ depends greatly on the base field. Considered as curves over the real numbers some equations such as y2 −y=y3−7 give rise to irreducible curves whereas others, for example, y2 +y=x3−x give reducible curves see Figure 5.2. As curves over the complex numbers, however, this is not the case. Similarly x2+y2=1 is a circle when considered over the real numbers (and hence is irreducible) over the but is the intersection of two lines complex numbers. A variant of ‘irreducibility’, called ‘absolute irreducibility’, is thus required which means irreducible when

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considered over an algebraic closure of the given base field or, in Weil’s language, irreducible over the universal domain. By considering only absolutely irreducible varieties Weil greatly simplifies his task and is able to remain close to the work of his predecessors (‘to finish, in harmony with the portions already existing
’), while use of the universal domain and consideration of cycles on the varieties allows him to deal effectively with the degenerate cases which were so troublesome to the older geometers. In summary we can say that Weil achieves the generality he requires by: • setting his objects in a context which is sufficiently large to permit all of the constructions that are required; • defining quite narrowly the principal objects of attention using the context given above; and • considering objects which do not fall under this narrow definition as separate kinds of object which are none the less made up of objects of the simpler type. There is a further subtlety in Weil’s approach which does not appear in the foregoing discussion. His definition of variety proceeds through a technical notion of ‘specialization’ of a field and involves an equivalence relation under this operation.24 A definition of variety is thus obtained which does not depend in the first instance on a field of definition (as the above rather simplified description may have led the reader to believe). However, since all of the constructions are performed inside the universal domain, a field of definition can always be found. This procedure also requires the further technical notion of a ‘regular’ field extension to ensure that the co-ordinate rings of the varieties do not acquire ‘divisors of zero’ when the base field is changed (e.g. when the variety in question is looked at over the universal domain). So defined, varieties have a natural notion of ‘generic’ point (roughly speaking, a point which satisfies only the conditions required to be a point of the variety and no other ‘special’ conditions) and one can proceed, in a fully rigorous fashion and without relying on the assistance of methods from complex function theory (which would not be valid in positive characteristic), to replicate the arguments about ‘general position’ that had been so favoured by earlier geometers. The bulk of Foundations of Algebraic Geometry is taken up with this programme, including a development of intersection theory on algebraic varieties which Weil required for his proof of the Riemann hypothesis. I shall now contrast Weil’s strategy with a very different approach to the problem of achieving generality in algebraic geometry, but a final word should be added in connection with the use to which Weil himself

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put his work in Foundations. The full proofs of the Riemann hypothesis for curves and related results appeared in two monographs in French, Sur les courbes algébriques et les variétés qui s’en déduisent, and Variétés algébriques et courbes algébriques.25 In these texts Weil again insists on the continuity of his ideas with those of his (mostly Italian) predecessors and further explains that the introduction of the general theory of abelian varieties, beyond the theory of Jacobians of curves, is necessary because of the light it sheds on the latter and the ‘mutual assistance’ the general theory and the more specific theory provide each other26. The list of loci where Weil insists on the importance of studying particular and general, the classical and the novel, side by side could be multiplied quite easily. 5.4 Grothendieck’s schemes and the Weil conjectures Following these efforts, Weil set out to apply his results on the Riemann hypothesis to a broader collection of equations than those defining curves over a finite field. His first success came with a very classical collection of equations related to so-called Gauss sums and led him to conjecture that the pattern of results which he had discovered for curves would hold true for all (higher dimensional) varieties over a finite field. In his detailed statement of this conjecture he referred to certain numbers associated to varieties which, in his view, were closely related to the Betti numbers and Euler-Poincaré characteristics defined by topologists. There was no obvious way, however, to extend these topological definitions, all of which presupposed notions of continuity, to the algebraic and ‘discrete’ realm of the varieties he was considering.27 Over the next 15 years, a concerted effort was made to adapt the growing arsenal of techniques from algebraic topology into algebraic geometry. Much of this work was done in Paris where both algebraic topologists and algebraic geometers were to be found. Although many ideas were put forward, by the late 1950s it was the ideas of Alexander Grothendieck that were beginning to command increasing attention.28 In the early 1960s Grothendieck and his school succeeded in defining in a purely algebraic way the topological notions required to attack Weil’s conjectures. The final result had to await P.Deligne’s work in the early 1970s but the foundations were in place by 1964. Grothendieck’s approach, while highly influenced by Weil, differs greatly from the work of his predecessor, particularly in connection with the aspects of the arguments designed to achieve generality and completeness. We have seen Weil argue for an ‘absolute’ universal domain whose choice will allow classical ideas to be given a rigorous foundation.

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Grothendieck instead argues for a ‘relative’ form of generality through various types of reflexivity, but at the price of introducing a completely new vocabulary and method of argumentation. Algebraic topologists (together with sympathetic algebraists) had introduced the notion of ‘categories’ into mathematics in an effort to formalize their study of the algebraic invariants that can be attached to topological spaces. Mathematical ‘objects’ were to be arranged in categories and related to one another by collections of mappings (‘morphisms’ or ‘arrows’)—Categories in turn could be related to one another by ‘functors’ and functors to one another by ‘natural transformations’. In this way the entire collection of topological spaces and their mappings could be related to a collection of algebraic objects and their mappings and the extent to which the structure of one category determined the structure of the other could be studied. This idea proved immensely powerful and spread quite quickly. Grothendieck found it ideal for his new vision of algebraic geometry.29 The crux of the matter is the simple observation that categories allow a great deal of ‘reflexivity’. For example, given a category consisting of objects and arrows between them we can reverse things and consider a new category whose objects are the arrows of the original one and whose arrows are the original objects. More significantly, the functors between categories can themselves be made into categories with the role of arrows being taken by natural transformations of functors. Grothendieck’s approach is to use these properties of categories to avoid picking ‘absolute’ starting points such as universal domains for his arguments and to replace the absolute starting points for his argument with reflexive or relative ones. We can see how Grothendieck’s argument works by considering what is meant by the ‘points’ of a geometric object such as an algebraic variety. It is clear that even simple equations defined over the rational numbers such as those given above have many ‘points’ on them (i.e. solutions) which are not defined over the rational numbers. Weil adopted the use of universal domains precisely to ensure that all of the fields defined by such points would be included in his structure before setting out. Grothendieck proceeds differently. He considers such equations to define a ‘scheme’. This can be thought of alternatively as an object which ‘has’ points defined over many fields or as a functor or ‘machine’ which, given a field (or more generally an algebra over a given field viewed as a base), defines a set, namely the set of points ‘on’ the scheme which are defined over the given field (i.e. a functor from the category of fields or algebras over a given base field, to the category of sets). Now a functor (from a category to the category of sets) which can be seen alternatively as a functor or an object (of the original category) is called a ‘representable functor’ and a scheme is merely a certain kind

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of representable functor.30 The shift of emphasis from objects such as varieties to representable functors, which at first seems largely linguistic, has far-reaching consequences. If we study this functor a bit more closely we learn that it has certain compatibility properties in that if a given algebra is ‘covered’ by a collection of algebras (in the case of schemes, a privileged notion of covering given by the algebraic procedure of ‘localization at a prime ideal’ is available to provide a notion of covering) then the set of points given by the functor over the original algebra is precisely the collection of points given by the functor on the ‘overlaps’31 of the covering algebras. A functor which behaves this way in respect of overlapping coverings is known as a ‘sheaf’ (in this case a sheaf of sets). The category of schemes therefore is a category of sheaves on algebras. We can go further if we are prepared to expand the notion of ‘covering’ used to describe the compatibility properties of schemes. A range of notions is in fact available from algebra (other than the localizations used for schemes) which give rise to different ‘topologies’. Correspondingly, a category of functors which are sheaves on algebras for such topologies is known as a topos.32 In this manner Grothendieck defines categories which are more ‘general’ than those of Weil’s varieties (in the sense that the category of varieties can be seen as a sub-category of the category of schemes) without relying on a given ‘absolute’ universal domain. The categorical approach allows one to deal with geometric objects alternately as objects of a category or as functor between categories. Hence Grothendieck replaces an absolute approach with a relative or reflexive one. For instance, the family of curves discussed above could be considered as a single scheme ‘over’ a ‘base scheme’ which included the parameter t. In this case it would have ‘relative’ dimension 1 since ‘over’ each value of the parameter one would find a curve. Alternatively it can be considered as a two-dimensional scheme over a ‘smaller’ base (i.e. one without the parameter). Grothendieck time and time again shows the power of the former point of view by proving theorems in a ‘relative’ context and showing how the usual ‘absolute’ version becomes a simple consequence.33 The scheme approach would not make distinctions in kind between the curves as irreducible objects and the cycles arising from the intersection with a line as Weil did. All are schemes, albeit with different properties, and no particular field need be chosen as the domain in which ‘all’ of the points of these schemes can potentially be found. The rings of functions which define schemes such as the point at (0, 0) in the equation y2=x3 do differ from the rings describing the same point on curves where t  0 in that they are not

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‘reduced’ in the technical language, but this does not prevent the point itself from being a (non-reduced) scheme!34 Weil’s universal domains are intended to specify the range of possible values that the variables in the equations defining geometric objects can take. Varieties are singled out as objects whose equations cannot be ‘factored’ when the coefficients are considered to belong to the universal domain (in contrast, for example, to the equation of the circle, considered as an equation with real coefficients which factored when it was considered as having complex coefficients). Grothendieck not only abolishes the universal domain, he also abolishes any fixed distinction between variable and coefficient. When one scheme is viewed relative to another, i.e. ‘over a base scheme’, the values of the variables are given by the base scheme. But this base is not viewed as fixed once and for all. Instead a process of ‘base change’ is established which allows systematic review of those properties which remain fixed and those which vary when one base scheme is substituted for another.35 Thus there is no distinction between coefficient and variable that can be fixed once and for all. This aspect of Grothendieck’s work was (and is) somewhat disconcerting for mathematicians coming to it for the first time, but is crucial to the innovations which he introduced. Grothendieck’s constructions are far from the ‘palaces’ which Weil suggests belong to algebraic geometers by birthright. Rather than refurbishing and renewing the old constructions he has instead created an entirely new type of architecture and in the process is forced to make extensive use of the ‘makeshift algebraic constructions’ Weil carefully sought to avoid placing reliance upon. Both Weil and Grothendieck seek generality in order to be able to analyse rigorously geometric situations to which the older algebraic geometers paid little attention. This generality permits consideration of ‘degenerate’ cases, objects with singularities, ‘non-reduced’ objects and the like. Whereas Weil sought this generality through a large ‘domain’ in which the results of his constructions could be contained, Grothendieck seeks generality by expanding the category of objects under consideration until a certain ‘self-sufficiency’ is acquired, i.e. until a natural reflexivity can be found so that the objects in the category can be related to each other and operations on them can be undertaken without going outside the original category. Furthermore, the objects do not require a predetermined ‘fixed point’ outside of the category for their specification.36 The notion of a representable functor is the key technical ingredient in finding this self-sufficiency. With these contrasting approaches to the problem of completeness of a mathematical subject matter in front of us we can now see how Euclid’s approach to the same issues relates to those of Weil and Grothendieck.

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5.5 Euclid’s approach to generality and closure Why does Euclid choose to end his treatise with three books on solid figures after ten books dealing with plane figures? In one sense there is no satisfactory answer to this question as we are too ignorant about both the structure of geometrical treatises in Euclid’s time and the ultimate purposes for which the Elements was compiled. The understanding of Euclid’s argument developed to this point does, however, allow us to put forward some hypotheses which in turn may clarify the role of these three final books in the overall plan of the treatise. In Book I the notion of figure was introduced to explore the indeterminate area left open by the definitions of bounded surfaces and plane surfaces. Figures are ‘that which can be defined’ because they are ‘that which is bounded’, but many figures (e.g. triangles) are defined without reference to an ambient ‘plane’ and there is no sense in which we can see the study of figures in Books I–X as somehow ‘adding up’ to a study of ‘the plane’. The figures are examined on their own and exist independently of any ambient space. The reason for this is simply that Euclid’s investigation operates precisely by leaving the ‘geometric properties of space’ undetermined.37 This point has been repeatedly emphasized in Part I. On the other hand, there is no notion of completeness in the investigation of plane figures of the sort which might arise from some attempt at classification of such figures in a determinate way or some other form of boundary or delimitation of the subject matter they constitute. The introduction of the new contexts of Books V–X does not alter this since they ‘only’ provide the means to compare previously defined things in new and more sophisticated ways. The reader is therefore entitled to (and indeed should) pose the question, ‘What does the investigation of Books I–X then add up to?’ The response to be proposed can be phrased in very simple terms, ‘Books I–X provide the materials for the construction and investigation of regular solid figures whose study does amount to a completed whole.’38 The Definitions of Book XI begin in a manner which resembles those of Book I in defining a solid and a bounded solid, but the resemblance stops there and holds only at a superficial level. As previously remarked, for Euclid there is no equivalent in the realm of solids to straight line and plane surface. If we consider the sequence ‘point, line, surface, solid’ we see that: • the point is all measure with nothing to be measured; • the straight line is both measure and measured;

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• the plane surface is both measure and measured but in distinct ways whose relationship is essentially indeterminate; and • the solid is only measured and does not measure as there is no straight line/plane surface equivalent. Solids provide a context in which the previously defined and studied materials can operate, not by permitting more extensive and incisive comparisons as in Books V–X through the introduction of ratio and proportion, but by being parts of a whole. Solids themselves cannot be parts of anything else since they have no properties of their own to provide, being entirely determined by their parts and not measuring anything else. This means that solids are a ‘neutral’ context which does not alter the properties of the figures which are their parts.39 This does not mean, however, that there is nothing to say about solids or that, in some sense, everything that can be said about solids can be deduced from appropriate properties of other figures. Indeed Book XIII is famous for its demonstration of the existence of the five regular solids and the proof that no others exist. It seems difficult to maintain that these famous theorems are merely corollaries of previously demonstrated facts about plane figures. Euclid achieves completeness by a kind of reflexivity in which figures become parts of other figures. But this reflexivity differs from Grothendieck’s process of studying schemes relative to one another in that there is a clear and fixed distinction between the parts and what the parts are of. On the other hand Euclid does not, as Weil’s argument might suggest, seek a fixed ambient space or figure in which all of his constructions can take place, although, as we shall see, the regular solids are constructed so as to be contained within spheres. Once again Euclid’s argument can be seen to thread a path between the positions taken by more recent mathematicians and may thus offer new mathematical directions. A more detailed examination of the contents of Book XI–XIII will bring out the specific nature of Euclid’s position. 5.6 Parts and wholes Given the important role played by the term ‘part’ in the analysis of the Elements up to this point it should come as no surprise that the transition to the final three books of the treatise is marked by yet another use for this many-valued concept. Since solid figures are entirely determined by the parts which constitute them it is plausible to refer to them as ‘wholes’. This would be inappropriate if solid figures in turn were to be made into constitutive parts of other things. That would give them an ‘active’ role whose nature would in turn reflect on their parts.

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The introduction of this new relationship of part and whole thus has a second aspect as well since one can distinguish between, say, the sides and angles of a triangle which are its defining or constitutive parts (the verbal definition of the figure runs in terms of these parts and these are the parts of the boundary or ‘geometric’ definition of the figure) and, say, a cone which forms part of a cylinder. In the latter case we have a quite general or loose sense of part and whole in which no relationship of definition or constitution is suggested. Up to Book XI Euclid has required the term ‘part’ either to tie his definitions to the things defined (Books I–IV) or to establish the contexts in which the things so defined can be related to each other (Books V–X). With the introduction of solids he no longer needs to use the notion of part either to define or measure figures and he can therefore introduce the more general meaning to the term. Of course in this looser sense solids can be parts of other figures, but this does not necessarily involve the relationships of measurement which have gone along with the notion of part in previous books. We can see this shift quite clearly in the manner in which Euclid handles the definitions of angle in Book XI and the introduction of figures which involve ‘more than one plane’. In Book I an angle, although it is made up by the lines which contain it, is not, strictly speaking, considered to be a ‘whole’ in the manner of a solid figure since there is a reciprocity between the angle, which helps to give determination to the lines which make it up, and the lines which determine it. Euclid defines ‘plane angle’ in Book I using the phrase ‘in one plane’ in his definition even though he has no means at that point for determining whether something is ‘in one plane’ or not. It is necessary, therefore, to consider how angles receive determination both by being placed into figures, as in Book I, and, now in Book XI, by being placed in a context in which multiple planes can be distinguished. Recall that plane angles were defined in Book I (Definition 8) as the inclination to each other of two lines in a plane which are not in one straight line. This definition was followed by the defintion of an angle contained by straight lines (Definition 9) and by the definitions of right, acute and obtuse angles (Definitions 10–12). In Book XI an inverse process is followed in which first the definitions of lines and planes at right angles to another plane are given (Definitions 3, 4), followed by the definition of the inclinations of lines and planes to a given plane (Definitions 5, 6) and only subsequently is the definition of solid angle given (Definition 11). The difference in procedure is due to the introduction of the new context of solids in which points, lines, planes and angles can all act as parts. The introduction of plane angles permitted the consideration of two lines of the same length as separate and different in spite of the fact that their essential attributes as lines did not differ. The notion of angle provided a specific context in which

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such discriminations could take place between two lines. Subsequently, with the introduction of figures with many parts, similar discriminations could operate between many lines although always with reference to a specific figure, hence to a specific number of lines. Solids generalize this process by providing a context in which many points, lines, angles and planes can be discriminated. Furthermore, as with the procedure in Book I, the introduction of these new contexts provides a new function for at least one of the types of parts which make them up. In the previous commentary on Book I in Part I, references in Euclid’s text to objects being ‘in one plane’ (most notably in connection with the definitions of angle and parallel lines) have been passed over in silence. It is time to rectify this omission. The subject matter of Book I does not contain circumstances in which plane surfaces which are the same as far as the terms used in their definition (namely their length and breadth) can be distinguished. Recall that points can be distinguished as they function in delimiting lines and lines can be distinguished as they function in containing angles; there is no analogous circumstance in Books I–X in which planes function so as to allow such distinctions. It is only with Definition 2 of Book XI that planes (among other surfaces) can be seen to function as the extremities of solids. In Book I, therefore, we cannot take the phrase ‘in one plane’ to rest upon or to imply any sense of ‘in one plane as opposed to another’. Euclid cannot provide any criterion for determining when two lines are in one plane in Book I and so, in a sense, the phrase ‘in one plane’ has only a hypothetical meaning (‘if it can be shown that two lines are in a plane’). Note, however, that this phrase also has a positive function in its uses in Book I in that, where it is used, that which is being bounded (or, in the case of parallel lines, for which the non-existence of a boundary is being stated) is a part of a plane (albeit an indeterminate one). Now, with the introduction of solid figures, the hypothetical status of these definitions is removed and indeed the first three Propositions of Book XI are precisely about ‘being in one plane’. As these Propositions have not been well received by commentators40 (and as they do not involve the later definitions in Book XI) it may be useful to review them briefly here. In order they assert that: • a part of a straight line cannot be in the plane of reference and a part in a plane more elevated; • if two straight lines cut one another they are in one plane and every triangle is in one plane; and • if two planes cut one another their common section is a straight line.

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Here we meet the new generalized sense of part as well as the differentiation of planes without reference to their defining attributes. Euclid speaks of ‘the plane of reference’ for the first time (the Greek is or ‘underlying’) but his intention is merely to distinguish two planes. Heath (among many others) gives this Proposition, its proof and the proofs of the other two Propositions short shrift. As he puts it, the proofs are unsatisfactory because ‘Euclid is unable to make use of his definition of “plane”’ in them. One is tempted to respond, ‘precisely!’. The absence of Postulates for the construction of planes has been noted previously and related to the fundamental issue in Euclid’s investigation in Book I, the indeterminate region between surfaces as bounded by lines and surfaces as measured by straight lines, the region occupied by figures. Planes are at bottom undetermined and do not possess the self-determination of straight lines, points, circles, right angles and the complexes of lines and angles that make up triangles. Planes are constructed and determined by other things, they cannot be known (one might even say ‘do not exist’) without reference to these other things. In particular, Euclid has no reference to an infinitely extended geometric object that might be called ‘The Plane’. His planes are constructed, bounded and determined by other things. In the above Propositions, the phrase ‘the plane of reference’ does not mean ‘an infinitely extended (or extendible) object “The Plane”, but rather some specific plane figure or plane determined by a boundary whose properties happen not to be relevant to the question at hand. What is important is not the boundary or definition of the plane but simply that there are (at least) two of them. The proof of Proposition 1 proceeds by contradiction using the Postulates to construct a hypothetical continuation of the line in the ‘plane of reference’ giving rise to a segment of a line which is putatively part of two lines (once again a generalized sense of part). This leads to a circle whose diameter cuts off unequal circumferences and a contradiction. The commentators focus on the ‘unwarranted assumption’ that one can extend the line in the ‘plane of reference’, noting of course that Postulate 2, allowing the extension of a straight line, has no reference to a single plane. But this objection only makes sense if one forgets that the Postulates operate by providing the exact data required to determine an outcome which then becomes self-determining. In particular, if one did not know, in connection with Postulate 2, whether the extended line were to be in one plane or another, one would need a means for determining this aspect of the outcome. As noted above, however, Euclid’s Book I operates ‘in one plane’ in the sense that no means for determining the differences between planes (apart from their definitions) is provided. Only with the introduction of solid figures in Book XI can we speak of multiple

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planes and refer to the objects of Book I as lying in a given plane (Euclid’s ‘plane of reference’) as opposed merely to being in a single plane. If, on the other hand, Euclid were unable to extend his line segment in the plane of reference he would require a further determination in the form of a Postulate to say how the extension is to take place. There are, however, no Postulates on which this further determination can rest in his system. Put another way, the unspecified or hypothetical ‘in one plane’ of Book I becomes ‘in a given plane’ or ‘in a single specified plane’ in Book XI. The proof of Proposition 2 is similar and uses a general notion of ‘part’ of a triangle. Heath objects that the ‘part’ referred to is one bounded by straight lines and that it might be possible to have a ‘part’ of a triangle bounded by other (e.g. curved) lines which was in one plane while another ‘part’ was in another plane. This is indeed the case. Euclid’s demonstration considers only particular types of figures constructed as parts of others. The generalized sense of part at work here does not extend to figures composed of lines which are not defined in Euclid’s system and ultimately rests on his notions of definition and delimitation which were discussed at length in Part 1. Finally, the proof of Proposition 3 takes the matter of ‘in one plane’ a step further by arguing that if the locus of intersection of two planes is not a straight line, then there will be straight lines in each of the two planes with the same extremities. Here Postulate 1 is called upon in the same manner as Postulate 2 in Proposition 1. In Book I there was no way to speak of multiple planes, hence the Postulate operates without the possibility of such determination. In Book XI this means that it operates ‘in one (given) plane’. Thus the new context of solids in Book XI both provides an expanded notion of part and allows us to distinguish planes ‘which are otherwise the same’ in their functioning as extremities of surfaces. The essential indeterminacy of planes, however, means that this new functioning is not as productive for our knowledge of planes as the functioning of lines as extremities of planes was for our knowledge of lines. The variety of plane figures means that we shall have many different boundings to contend with as opposed to the single type of bounding given by straight lines and this is reflected in the various modes of definition used in Book XI. To return to the definitions of angle in Book XI we now see that angles are not used here as a context to give new functionings to their constitutent parts as in Book I but as complex wholes, made up of parts which bear relationships to one another which can be isolated and spelled out. Euclid still takes equality as the principle of measurement and so defines ‘right angles’ first and then, more generally, ‘inclinations’ in terms of right angles but angles are now between line and plane and

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plane and plane as well as between line and line. The key point here is the establishment of contexts in which multiple planes are distinguished and planes are related to planes, lines related to planes and lines related to lines both within and between planes. The Definition of a solid angle (Definition 11) culminates this process by defining such an angle in terms of either (sic!) the inclination of lines which are not in a single plane to one another or an angle contained by two or more plane angles (which are not in a single plane). The alternative in the Definition shows clearly that the intimate relationship between parts and what they constitute in the Definitions of Book I has been left behind here. A great deal is already known about lines and angles by this point in the argument and, in a sense, we shall not learn much more about them by placing them in this new context. We shall however, be able to make distinctions which were not previously possible between things in different planes and, of course, we shall be constituting some interesting objects, solid figures. 5.7 Solid figures as complex wholes In Book I we saw that definition and limitation were identified through the notion of figure and that the parts of figures were the parts of their boundaries, i.e. the parts of their definitions, and these consisted of lines and angles. The boundaries of solid figures are much more various, consisting of lines, angles between lines and planes, planes of various types and their parts and finally other surfaces. While it is still the case that a figure is defined by its boundary and that the definable is equated to the bounded in Book XI, the nature of Euclid’s investigation has altered. The definitions do not reveal a subject matter to be explored since there is no equivalent of the notion of a plane here. Instead a series of wholes is established into which the materials of Books I–X can be placed as parts and Euclid’s investigation turns to the construction of these complex wholes, their measurement and the demonstration that they exhaust the possibilities available. A pair of comparisons with Book I will help to bring this out. In Book I the definition of parallel lines served as a type of limit on the subject matter to be explored by negating limits. Parallel lines and planes were made coterminous, reciprocally measuring each other, and the negation of limit provided a limit for the set of definitions and the subject matter they outlined. In Book XI Euclid refers again in a shorthand way to the formula of lines which do not meet in his definition of parallel planes, but with two differences. First there is no ambient in which the planes are contained. There are no possibilities for kinds of solid spaces as there are for straight lines and plane surfaces

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so the parallel planes do not act to measure anything the way parallel lines ‘measured’ planes. Second the definition of parallel planes does not close the set of definitions but sits in the middle of the sequence. The sense of closure is provided instead by the definitions of the five types of regular solid which, as will be demonstrated, exhaust the possibilities. The subject matter here consists of figures which are complex wholes, made up of other figures but without the reciprocal sense of bounding and bounded that permeated Book I. Regularity is specified not by the equivalent of straight lines or plane surfaces which are regular as ‘measured things’ but rather by regular solid figures which are such by virtue of the regularity of their parts. The differences in the nature of the definitions of Book I and Book XI can also be seen quite clearly in Definition 10 which states that ‘Equal and similar solid figures are those contained by similar planes equal in magnitude and multitude’. There has been much scholarly debate concerning this definition with many commentators seeing it as a (false) proposition rather than a definition. No definition of equal figure (or equal anything) was given in Books I–IV and the implications of Euclid’s use of ‘equality’ as a principle rather than a consequence of measurement in the Elements have been stressed repeatedly. In Book V Euclid gave a definition of ‘in the same ratio’ and it was noted that this implied that ratio was not a quantity. Here we have a third variation on the theme. By Definition 2 of Book XI we know that surfaces are the extremities of solids, but we have no Definition to tell us what the surfaces contain when they are the extremities of something. By Definition 14 of Book I, the surfaces contain a figure, but a figure is not a measurable like length or breadth or depth. Recall that the equality of triangles in Book I is carefully distinguished from the equality of their parts and the notion of the ‘area’ contained by a triangle is also carefully distinguished and developed (cf. the commentary in Part I on Propositions 4, 8 and 37−44). In defining equal solid figures in Book XI Euclid therefore cannot be asserting the equality of their ‘volumes’ (this of course would be a proposition, not a definition), in fact he is merely providing a meaning for equality of two complex wholes. The discussion of the measurement of solids in Books XI–XIII in no way depends on this verbal convention and in many ways goes far beyond the concepts that it embodies. The reader will no doubt find many other examples of the differences between Book I and Book XI which illustrate the shift that has taken place in the nature of Euclid’s investigation over the course of the treatise. Of interest to us here are the means by which Euclid gets completeness and closure in his science through the introduction of solid figures. A brief examination of the argument in the Propositions of Books XI–XIII will provide further insight into these questions.

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5.8 Constructing and measuring solids The solid figures introduced in Book XI not only allow Euclid to introduce the grammar of plural planes but also provide examples of figures which, as it were, provide their own measurement. The sequence of Propositions in Book I led up to Proposition 47 on right-angled triangles exhibiting a figure whose definition was sufficient to entail a relationship of measurement between its parts. Book III, on circles, which has not been discussed previously, extends this idea by showing that circles provide a context of equality in which relationships can be established between parts of figures and between different figures. This abstract context is then made concrete in Book IV by the introduction of the notions of circumscription and inscription of figures by and within circles so that one can say that circles provide circumstances under which measurements can be made. Of course at that stage in the Elements, measurement consists simply of relationships of equality and greater than/less than. With the introduction of the more complex notions of comparison and relationship in Book V, all of this can be extended from equality to similarity, and this is to some extent carried out in Book VI. The figures which make up the boundaries of solid figures therefore come with a set of relationships which we may term ‘measurements’ and these measurements can be used to set up relationships between solid figures as well. The procedure of the earlier books is reversed in Books XI–XIII as figures with determinate relationships amongst their parts are used first to define and then to construct complex figures which contain the simpler figures as parts. In these constructions there are relationships of measurement between all of the parts and between the parts and the wholes. In Book IV regular polygons are constructed in the context of circles by constructing them as circumscribed about and inscribed in circles. Either the circle or the figure can be taken as given and the process runs from what is given to the construction of a figure with certain specified properties. Book XIII employs a similar technique, but vastly expanded in the nature and precision of the comparisons which can be made. In Book IV, comparisons can be made between the two figures in a pair of ‘circumscribed about/inscribed within’ but it is not possible to compare ‘pairs of pairs’. Book V remedies this but leaves open the question of whether and how arbitrary pairs can be compared. These questions are addressed in Book X. Books XI–XIII use all of this to:

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• compare figures of the same type which are inscribed within/ circumscribed about different but similar figures; • compare the parts of the figures in a pair inscribed within/ circumscribed about; and • compare the parts of different figures inscribed within/ circumscribed about the same figure. Spheres, like circles, are determined by a single part which facilitates all of these comparisons. Solid figures are thus constructed in a manner which brings about their measurement as well. Not intrinsically, as was the case with right triangles, but extrinsically, in terms of spheres in which they are inscribed. All of the means of measurement and comparison developed in the previous books and which were developed in a wide range of contexts can be put to use in considering a single solid figure! Conversely, the complexity of the comparisons requires all of the precision of Book X and its various types of incommensurables to achieve expression. 5.9 Summary of books XI–XIII The books dealing with solid figures in the Elements provide a generalization of what has come before in three different senses. First, the term ‘part’ is generalized from its previous roles to any relationship of part and whole. This step can be taken because solid figures provide ‘wholes’ for the parts that constitute them in a manner which distinguishes them from the figures of Books I–X. The figures of Books I–X have a double nature as congeries of parts and as examples of boundings of plane surfaces. Solid figures are merely congeries of parts since there is no equivalent of plane surface for solids. These wholes do not in turn become the constituent or defining parts of other wholes. Paradoxically, this means we can now distinguish between ‘constituent’ parts and other part/whole relationships and thus broaden the sense of part to include examples of the latter which previously could not be considered. Euclid is no longer relying on the strict relationships of part to part which permitted the definition of plane figures in Book I or the extension of comparisons between figures in Book V. The parts which make up solid figures are already determined so he does not need the notion of part for this purpose. Second, in the context of solid figures it is possible to distinguish plane figures whose definition and characteristics would otherwise not support distinction. The grammar of plane figures is altered by introduction of a plane of reference in which they may be supposed to lie and which may be distinguished from other planes. But the

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introduction of this new grammar differs from the introduction of a notion of plural points in Definition 2 or a plural of ‘straight lines of the same length’ in Definition 9. In spite of the extent of Euclid’s investigations in Books I–X, these do not and cannot constitute a complete study or ‘classification’ of plane surfaces. Book IV terminates with the construction of a 15-gon and points to the infinite possibilties for inscribing polygons in circles. As far as plane figures are concerned, the Euclidean notion of mathematical science is not one in which one can, as it were, circumscribe the subject matter. The definition of figure, as discussed in Part 1, does not support a general study of the properties of plane figures but rather opens up the possibility of defining and investigating certain particular examples. Surfaces and plane surfaces remain, in general, a mystery in the Elements although a great deal can be learnt about specific cases. Finally, the plane figures can become the material, the parts, of solid figures. Since solid figures are thus made up of figures which already support precise comparisons it is possible to set up circumstances in which solid figures and their parts are related in all of the ways in which figures have previously been seen to measure each other. In this sense solid figures are the general context of measurement incorporating all that has gone before into a single example or set of examples. These complex relationships between parts and wholes circumscribe not the measurements but the possible figures which can support them. Thus from Euclid’s point of view although there is no single general study of plane figures there are a limited number of solid figures whose parts and their relationships embody all of the types of comparison and measurement that one can make with plane figures. The study of plane geometry does not constitute a whole and cannot achieve completion; the study of solid figures in which plane figures function as boundaries and parts is a study of wholes and arrives at a natural conclusion.

6 Generality in contemporary mathematics

6.1 Some contemporary problems Much has been made of the penchant of modern mathematics for the general and the abstract.1 From Hilbert to Grothendieck many of the great movements of mathematics in this century have been dominated by attempts to increase the scope and range of mathematical subject matters by increased abstraction and generalization. While it may be argued that the mathematics of the past twenty or so years shows a reaction to this tendency, the flow of generalization and the desire to deal with cases previously considered inaccessible due, for example, to the presence of singularities or absence of restriction to finite dimensions has, if anything, intensified. Weil and Grothendieck exemplify opposite approaches to the problems faced by mathematicians seeking both to extend existing subject matters to new areas and to find the natural boundaries or limitations of their new domains. As noted above, they both exhibit areas in which their approach overlaps Euclid’s and areas where it diverges. It would be possible, for example, to compare in detail Euclid’s use of spheres to circumscribe and measure his figures with Weil’s use of universal domains in which all algebro-geometric constructions are to be made. Equally one could consider the reflexivity that Euclid finds in making figures parts of other figures with Grothendieck’s relativization of algebraic geometry through the notion of a representable functor. Rather than continuing in this vein, however, to close the argument of this book I shall deviate somewhat from the pattern employed to this point to focus on a pair of current mathematical issues rather than (directly) studying texts. The issues selected illustrate particular instances where these two contemporary mathematicians have sought to extend the boundaries of existing subject matters with a view to enclosing in a single study certain disparate phenomena which display tantalizing but as yet unfathomed

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similarities. My purpose is to suggest how the example of Euclid’s approach to generalization and completeness and the textual and argumentative strategies discussed above may be related to issues in contemporary mathematics. The problems to be considered (very briefly) are Weil’s suggestions on including the ‘infinite prime’ in arithmetic algebraic geometry, which today mostly travel under the rubric of Arakelov Geometry, and Grothendieck’s search for a categorical framework in which to study homotopy theory as outlined in his unpublished ‘A la poursuite des champs’. 6.2 Arithmetic algebraic geometry It is perhaps only a slight exaggeration to say that behind much of Weil’s scientific work we can find the haunting figures of Dirichlet and Riemann. Together these nineteenth-century mathematicians opened up new vistas in number theory by introducing analytic and geometric techniques from complex function theory. While Weil never seems to have felt ‘philosophical’ scruples on this account, he has often given expression to an exceptionally strong drive to understand the origins and principles of the application of such methods in number theory.2 In 1938 and 1939 he published a series of three articles dealing with the analogies between number fields, fields of algebraic functions in positive characteristic and fields of functions on complex algebraic varieties viewed from the point of view of differential geometry. These articles had as their central theme the well-known Riemann—Roch theorem which Weil explored from a variety of angles producing ideas which today must be considered to be among his most visionary.3 In the half-century since their publication these articles and the ideas they contain have forced mathematicians to contemplate the possibility of introducing new objects into mathematics, the study of whose properties requires a combination of algebra, arithmetic and geometry. The analogy between number fields and function fields (whether in positive characteristic or over the complex numbers—in any event the discussion below is limited to the case of function fields of a single variable) is based on the concept of ‘place’, a notion which evolved slowly but naturally from Kronecker’s ‘generalized arithmetic’.4 From this point of view, a number or a point on a curve (or other variety) is not in itself a value but rather something which gives values to functions, i.e. a ‘place’ at which a function can be evaluated. One considers, for example, an integer as being a function which is ‘evaluated at’ the various prime numbers, receiving the value 1 ‘at the prime p’ if it is not divisible by p (which will be true for all but a finite number of primes) and the value 1/pn if it is divisble by pn. This can be

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extended to rational numbers by considering numerator and denominator separately and then combining the results. One can then ‘complete’ the system by adding in all infinite sequences of rationals (Cauchy sequences) using the above valuation rather than the usual notion of ‘absolute value’ to govern convergence of a sequence. In this way one arrives at the system of p-adic numbers and there is one such system for each prime. It is possible to mimic this process in an arbitrary number field as well. The analogy to geometry comes about by considering the points of a geometric object, e.g. the complex plane, as ‘places’ where functions are evaluated. The point plays the role of the prime as described above and a function is ‘divisible’ by a point if it vanishes at the point. Just as in the case of the p-adic numbers this gives a ‘valuation’ at the point to the field of functions which can be ‘completed’ in a manner wholly parallel to the arithmetic case and, just as in that case the process was not limited to the rational numbers, so to here it can be extended to the more complicated curves of higher genus. There is no limitation here on the base field so that we may consider functions fields over the complex numbers or over finite fields. Weil notes that the analogy is useful in that it brings out differences as well as similarities. In the case of function fields one typically does not consider objects such as the complex plane because they are not ‘complete’, but rather one adds a ‘point at infinity’ to complete the plane and arrive at what is sometimes called the Riemann sphere. This process of completion goes back to the early days of projective geometry and became embedded in the study of algebraic geometry in the nineteenth century. One of the many advantages of such complete objects is that algebraic functions on them display a very useful property. An algebraic function can be considered to map a given variety to the Riemann sphere and if the variety is a complex curve such a function will ‘hit’ each of the points on the sphere only a finite number of times and it will take the value 0 as often as it takes the value . Following the analogy above, a function which takes the value zero at a point is ‘divisible’ by that point and the order of the zero is the ‘power’ to which the point ‘divides’ the function. Similarly, a function which takes the value  at a point is such that its reciprocal is divisible by the point. Let us agree then to give the function the value n at a point P if it has a zero of order n there, the value −n if it has ‘an ’ of order n there (usually referred to as a ‘pole’ of order n) and the value 0 otherwise. The useful property referred to above is the simple fact that the sum of such values for any algebraic function over all points on a complete curve is 0 (note that a given function or its reciprocal will be divisible by only a finite number of points just as a given integer is divisible by only a finite number of primes).

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To find the analogy in the arithmetic case we must multiply the values of an integer at each prime according to the recipe given above (only a finite number of these numbers will differ from 1 so the infinite product makes sense). If the result were to follow the case of the values of a function on a curve we would expect this product always to equal 1. This, however, is not the case. If we consider the number 6, for example, we see that it has the value 1 at all primes except 2 and 3 where it has the values ½ and respectively, giving our product the value . The problem is that we have no equivalent of the ‘point at infinity’ to balance the values at the ‘finite primes’. To arrive at such a result we would need to reinstate the usual ‘absolute value’ as a ‘prime’ (technically in this case a ‘real infinite’ prime, if we had started with a ring of algebraic integers such as Z[i] we would find ‘complex infinite primes’ as well) and to ‘complete’ our number field with respect to this prime as we did in the case of the p-adic numbers. The result in this case are the familiar real numbers! Now if we take the ‘value’ of 6 at the and the ‘real prime’ to be its usual absolute value then we get analogy with the function field case is reinstated. It was Weil’s insight that the introduction of such infinite primes is not a formal matter of ‘balancing the books’ or merely mimicking wellknown results in other areas, but rather that the analogy between number fields and functions fields, by revealing this difference, both justifies and requires the introduction of the tools and techniques of analysis into number theory. This leads directly to the notion that the proper subject matter for number theory is not number fields seen merely as extensions of the rational numbers but number fields as they can be completed at all primes, both the finite and the infinite ones. In this way Weil found a natural explanation for the introduction of tools from complex analysis into number theory as practised by Dirichlet and Riemann. 6.3 Putting the pieces together The first technique for combining these completions together was the theory of the adèles. These are a type of product of all of the completions of a given number field (infinite primes included). Weil championed their use over many years and allowed himself to suggest that they might be superior as an approach to arithmetic problems in algebraic geometry than Grothendieck’s schemes. The adèles can be seen as generalizations of more traditional objects such as topological groups or rings but with significant alterations due to the fact that the product involved must be taken over an infinite number of primes (the product in fact is ‘restricted’ so that this problem is more apparent than real). There can be no doubt that the introduction of this new object and with

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it the ability to import wholesale into number theory the tools of analysis had dramatic effects and they have since become a tool that every number theorist needs to master.5 In recent years, however, the force of Weil’s analogies and the ideas behind them have been rediscovered and have led to the introduction into number theory of other new types of object known as Arakelov varieties. It was the completion of the ‘trilingual text’ or ‘Rosetta stone’ in ‘parallel columns’ consisting of number fields, function fields and classical Riemannian geometry which Weil insists was the driving force in his work. Quite apart from the more strictly algebraic tendencies of his German colleagues, he sought to establish the various connections between these subjects and, where necessary, fill in the gaps in one which appeared as a result of its being compared with the others. In the last twenty years mathematicians have sought to push these ideas forward through the creation of a type of geometric object which is both an algebraic variety defined over a number field and a manifold in the sense of differential geometry over the complex numbers. The role of valuations and places is supplemented by the notion of a (hermitian) Riemannian metric, a basic element of differential geometry, when the object is viewed as a (complex) manifold. Together the valuations at the ‘finite primes’ and this metric provide a ‘complete’ picture of the values of a function at a point, providing the ‘valuation at infinity’ that was missing above. The efforts of many mathematicians to develop this theory have come to fruition with the recent proof of a Riemann-Roch type theorem for these objects, a development very much in keeping with the spirit of Weil’s pioneering work of fifty years ago.6 One can also see strong analogies with Euclid’s use of figures as parts of other figures in this creation of a general arithmetic-geometric object containing ‘parts’ defined over a vast range of base fields. There remain, however, certain foundational issues to be clarified in the theory of Arakelov varieties, issues where it is perhaps not out of place to suggest that the analysis of Books XI–XIII of the Elements may be of some utility. The parts that constitute an Arakelov variety, i.e. the algebraic variety over a number field (or over the various local fields associated with the number field) and the complex manifold, are in a sense linked together only by the fiat of definition. This is to be contrasted with the manner in which the parts of a solid figure function together to constitute the whole. Consider Definition 1 of Book XI which defines a solid as that which has ‘length and breadth and depth’ before going on to say how it is bounded or defining the various figures. Although I have insisted on the importance of the fact that the equivalent of straight line or plane surface does not exist for solids, this of course does not mean that no definition at all of solid is provided. Given the fact that Euclid has very

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limited means to determine the measurement of solids through the definitions, the reader may be tempted to see Definition 1 of Book XI as somewhat formal. That this is not so can be seen by comparing Euclid’s approach with that used today in defining Arakelov varieties. As one has no a priori understanding of the ‘types of quantities’ involved in combining number fields and complex manifolds, one simply yokes them together as parts of an otherwise undefined whole and then proceeds to the investigation. Of course modern mathematicians will not be defining ‘solids’ or anything remotely similar here, but one can ask for a definition of the category of Arakelov varieties and more particularly of the maps or morphisms which exist between them. To date this does not appear to have been done and we remain in the dark as to the type of ‘whole’ which the parts of these fascinating objects compose. Weil does not shed much, light on the answer to this question but he leaves little doubt as to his vision of the consequences of finding such a whole. In a wonderful passage contained in a letter written to his sister, Simone Weil, from ‘Bonne Nouvelle’ prison where he was held for alleged draft evasion and desertion, Weil includes the following remarks on the progress from the analogies between the columns of his ‘Rosetta stone’ to science: If we look at this analogy more closely we find that once it affords us the ability to translate a demonstration from one theory to the next without alteration it has, to this extent, lost its fertility. And once the two theories have been amalgamated in a natural way the analogy becomes entirely sterile
The analogy has disappeared, as indeed have the two theories; gone as well as the troubling and delightful reflections of the one in the other, furtive kisses, inexplicable quarrels. What’s left is but a single theory whose majestic beauty leaves us cold. Nothing is more fruitful than these somewhat adulterous contacts and nothing gives more pleasure to the connoisseur
7 For Weil the coming of understanding is the bringing together of the two theories in a single theory, the realization that they are but two aspects of the same thing. In the process, of course, the separate identities are lost and with them the strange and wonderful sense of similarity and difference which attracts and perplexes the investigator. The single science which engulfs the two previous theories is, like the universal domain, large enough to incorporate or encompass all of the varied manifestations in which the phenomena it investigates may appear. Such a single theory has yet to emerge for Arakelov varieties

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and its construction is not a simple matter of reflection on the analogies and differences between the parts which constitute them. Weil expresses regret at the passing of analogy into a single science and thus regret at the passing from illusion and mystery to knowledge.8 We are entitled to ask at this point whether a scientist can rest content with a view of his own activity which is both depressing and dehumanizing. Surely this sadness at the passage from analogy to synthesis is related to a concept of mathematical whole which is so all-embracing and all-encompassing that it leaves no place for the individuality of the parts which constitute it. But must we accept that mathematical wholes are of the universal all-embracing kind that Weil espouses (reluctantly and without foregoing his attachment to adulterous contacts)? In contrast, Euclid’s wholes are not universal domains in which all previously defined and investigated figures are incorporated but simply examples of solids which can be seen as wholes and which are constituted by other figures as parts. The prior books of the Elements are not somehow superseded or absorbed by the more general notions of Books XI–XIII. The previous material retains its individuality but finds new employment as parts of wholes. If a theory embracing both algebraic geometry and differential geometry seems difficult to identify, perhaps a beginning can be found by creating certain specific wholes which do have a definition (other than as the sum of their parts) and in which at least some of the parts that are thought to go into making up Arakelov varieties can find a place. 6.4 Homotopy theory of categories In direct contrast to Weil, Grothendieck’s wholes are of a relative rather than absolute sort. Their universality resides not in a given fixed container in which everything takes its place but in a set of objects whose properties emerge because they can be shifted from one relative context to the next. By the simple process of determining when a functor is ‘representable’ Grothendieck shows how objects and operations can be interchanged so that distinctions between passive and active, variable and function, extension and base can be unfrozen to be rediscovered within particular circumstances rather than imposed from without. Categories provide the natural framework for the development of these ideas precisely because they do not depend on traditional systems of mathematical operations but rather schematize mathematical systems into objects and the arrows or maps which relate them. The success of the ‘language of schemes’ in modern algebraic geometry speaks for itself and many mathematicians are now comfortable with notions of ‘Grothendieck topologies’ and related

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concepts which, not too long ago, were considered to be of a very abstruse nature. Perhaps less well known are some of the other ways in which categories can be seen to provide ‘topological’ information. The effective cessation of his publications since the early 1970s leaves us with only the faintest of clues as to how Grothendieck might have pursued these ideas, but in widely circulated (though technically unpublished) notes and other writings some guidance for our reflections has been provided. The relative nature of categorical thought leads naturally to a kind of hierarchy of categories. If one considers a collection of categories itself as a category one has, in the first instance, the categories themselves as objects and functors between them as morphisms. But one also has maps between functors, so-called natural transformations as well. Hence, the collection of maps between the objects of our category of categories can itself be considered as a category—objects are functors and maps are natural transformations. Such a structure in which the arrows between objects (known as ‘hom sets’) have the structure of a category is a 2-category. Only imagination and perserverance are required to define n-categories!9 Curiously a rather natural use for these strange sounding things can be found. Starting with its introduction in topology in the 1930s the notion of fibre space has been widely used in many branches of mathematics. One of its most familiar forms is that of vector bundle, an object which is now at home in topology, differential geometry and algebraic geometry. Such an object consists of a pair of spaces, known as the total space and base space respectively, and a map from the total space to the base space which has the property that the inverse image of a point in the base space (such an inverse image is known as the ‘fibre’ over the point in the base space) is a linear space (vector space) and all fibres are linear spaces of the same dimension. Depending on the context in which one is working one requires that the map be continuous or satisfy some smoothness or other similar condition. Finally one also requires that every point of the base space be located in a small (open) set (or ‘neighbourhood’) whose inverse image in the total space can be represented simply as a product of that small neighbourhood and the fibre.10 Hence one says that ‘locally’ the total space is exhibited as a product over the base but ‘globally’ this may not be the case. The most familiar example of this phenomenon is that of the Möbius band (total space) which is locally the product of a circle (base space) and a line segment (fibre) but is not globally so.11 Grothendieck showed how this type of construction could be effected in purely categorical language with so-called ‘fibred categories’. These consist of a pair of categories and , a functor F between them and a certain kind of structure on F−1 (D) (the objects in such that F(C)=D),

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the inverse image of an object in . Namely one considers the category (F, D) whose objects are pairs consisting of an object in and an arrow v in such that v: F(C) → D. Arrows in this category, say from the pair (C, v) to the pair (C
, v
) are given by arrows in , w: C→C
, C
, such that v=v
[F(w)] .12 It is easy to construct a functor from the inverse image F −1(D) to (F, D), one sends C such that F(C)=D to the pair (C, id) (id being as the identity map id: D →D). The structure required to exhibit by F’ is an ‘inverse’ to this procedure (technically an ‘fibred over ‘adjoint functor’) which gives for any object in (F, D) an object in F−1(D). With this structure (F, D) can be seen as a shifted or translated version of F−1 (D) and the existence of the ‘inverse’ procedure suffices to ensure that the fibres above various objects of are ‘the same’ (technically one needs to consider as well the ‘dual’ category to (F, D) which is defined similarly except the arrows v go in the opposite direction, v: D→ F(C)). To summarize, above every object D of the category we find one and the same category as the ‘fibre’; this recovers the essence of the fibre space. We have also found a large number of categories ‘within’ the single category so that if we take the collection of fibre categories (C, v) we obtain a new category which starts from the same objects as but which incorporates the fibring structure. This collection of categories is a 2-category and so we see that 2-categories give, at least in this simple example, a version of fibre spaces. This is another illustration of the relative nature of categorical thought since the category can be considered either as a single category in itself or a 2-category fibred over by F and containing a series of categories within itself, namely the fibres of F. By specifying the situation further we approach Grothendieck’s notion of ‘stack’ (or ‘champs’).13 If the fibre categories are in fact ‘groupoids’, i.e. categories all of whose arrows are isomorphisms, then we have a ‘stack’. In Grothendieck’s formulation of algebraic geometry through representable functors stacks play a crucial role, for many of the most important spaces studied by geometers are defined by functors which cannot be represented by schemes. This is the case, for example, with the moduli spaces of curves which, in a sense, parametrize the isomorphism classes of algebraic curves. The functor which defines the so-called fine moduli space of curves cannot be represented by a scheme because the existence of curves with non-trivial automorphisms prevents the functor from having the ‘overlap’ condition described in the discussion of schemes above. This moduli space, however, is represented by a stack and, although their basic theory is much less familiar than that of schemes, it would appear that the stack framework is gradually gaining ground as the appropriate setting for the analysis of various moduli problems.

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6.5 Non-abelian homological algebra The categorical framework is capable of taking us much further however. One of Grothendieck’s signal early accomplishments on switching from functional analysis to algebraic geometry was the setting up of a system of homological algebra which permitted the wholesale importation of cohomological techniques from algebraic topology into algebraic geometry. Following his well-known ‘Tohôku’ paper of 1957,14 in which this work was set out, Grothendieck proceeded to extend the techniques of homological algebra in the direction of homotopy theory which is much more difficult than cohomology to describe in purely algebraic terms. His first accomplishment in this direction was the development, together with J.L. Verdier, of the notion of ‘derived category’. Such derived categories do not posess the internal structures familiar from the ‘abelian’ categories of homological algebra which are closely related to the structure of categories of modules over a commutative ring. At about the same time topologists also became interested in such exotic categories as they attempted to find algebraic and categorical models of homotopical phenomena. In this work one of the guiding threads is the notion of a ‘simplicial’ object. Starting with a single point one can sequentially create simple objects of higher and higher dimension in the following manner. To the point one adds another point and a line between them to get a line segment. To this line segment one adds another (non-collinear) point and two line segments between the new point and the old ones to get a triangle. To the triangle one adds another (non-coplanar) point and three line segments between the new point and the (three) old ones to get a pyramid. The process continues. The object so created is considered to have sub-objects of all dimensions and these are related to each by a series of easily defined maps. One can include, for example, a line segment in a triangle in three ways since one has three ways of picking two vertices of the triangle to correspond to the endpoints of the line segment. In a similar fashion every (n − 1)-dimensional sub-object can be included in n ways in an n-dimensional sub-object. Conversely we can take a triangle and obtain a line segment by removing, in one of three ways, a vertex; and similarly in n dimensions. These maps are known as degeneracy and face maps respectively and an object which has sub-objects in all dimensions and degeneracy and face maps between the dimensions (satisfying certain further conditions as well) is known as a simplicial object. We have used the example of points, lines, planes and so on to describe the so-called standard simplicial object but the same process works in any category and one can speak of simplicial objects in any category (simplicial sets and simplicial

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groups are probably the favourites).15 The categories of simplicial objects are far from being abelian and have been successfully used to model topological phenomena. The links between categories and topological spaces are provided by two further notions: the first is the topological realization of a simplicial object, the second is the nerve of a category. The construction of the ‘topological realization’ of a simplicial object is due to J.Milnor. This construction associates a topological space to a simplicial object and sets up the possibility of relating the category of simplicial objects (say sets for simplicity) with the category of topological spaces. On the other hand, Grothendieck and G.Segal have defined the nerve of a category. This construction associates to any category a simplicial object called its nerve, whose nth dimensional part consists of chains of n+1 objects and n arrows in the category:

Degeneracy and face maps are obtained by either extending an n-chain into an (n+1)-chain by repetition at the jth location or shrinking an (n +1)-chain to an n-chain by omitting the jth link. In this way one obtains a simplicial object. Taking its topological realization then gives a topological space (known as the classifying space of the category).16 This provides a simple way to go from categories to topological spaces. The difficulty is that the morphisms between categories or simplicial objects do not account for all of the maps between their realizations (the realization functor from simplicial sets to topological spaces is not ‘full’). How then to use these categorical constructions as models for topological phenomena? In the 1960s D.Quillen was able to show that certain non-abelian categories, including the categories of simplicial objects, can none the less be provided with a structure which allows one to imitate the constructions of homological algebra.17 As part of this work Quillen showed that the category of simplicial sets can be understood to provide an algebraic model of the category of topological spaces considered ‘up to homotopy equivalence’.18 This ‘homotopical algebra’ has provided very powerful algebraic models of homotopical phenomena. But Grothendieck has somewhat more in mind as he explains in his unpublished notes entitled ‘Esquisse d’un programme’. Homotopy theory analyses a topological space by studying its ‘Postnikov tower’, a series of fibre spaces approximating at each level the homotopy properties of the given space up to a certain order. Since the classifying space construction associates to each category a topological space it may well be possible to analyse an arbitrary category in a similar but purely categorical way with the stages in the tower being occupied by

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n-categories, each of which is fibred over the (n – 1)-category below it.19 The theory of stacks and more generally of the cotangent complex of a morphism developed by L.Illusie, one of Grothendieck’s students, can be seen as a first step in constructing this tower. An example of a small portion of the next step has been provided more recently by P.Deligne, who shows how ‘Picard’ categories may be viewed as an instance of Grothendieck’s categorical Postnikov tower in which only stages 1 and 2 are occupied.20 This can briefly be described as follows. Picard categories are groupoids, i.e. categories in which every arrow is invertible, which also have an operation (labelled ‘+’ for convenience) which is such that the functor +X for each object of the category is in fact an equivalence of the category with itself (translated by X so to speak). These axioms in turn imply the existence of a 0 object and can be supplemented by constraints which provide associativity and commutativity of + if desired. Given a category 21 a Picard groupoid over consists of data providing: • for each arrow f:X→Y in a category [ø] := Px,y in which every arrow is invertible; • for each triple of objects (X, Y, Z) in a composition o: Pz,y × Py,x → Pz,x such that Xo or oX give equivalences; • an associativity constraint for o which is compatible with composition in the sense that for ø1, ø2 in there are isomorphisms [ø1oø2] [ø1] o [ø2] and [ø3oø2oø1] [ø3oø2] o [ø1]etc. To complete the construction one notes that the collection of such data forms a category and this implies (by a standard construction) that there is a ‘universal’ Picard groupoid for the original category. Deligne shows that such groupoids give information not only about the (homotopy classes of) loops in the nerve of the category but also about (homotopy classes of) homotopies of loops (i.e. stages 1 and 2 of the Postinikov tower). Such a Picard groupoid is a category fibred over the original and hence an example of a 2-category construction. category Grothendieck has in mind the possibility of extending this type of construction to n-categories to provide a categorical model of all of the homotopy information given by the Postnikov tower and speaks of -groupoids, a type of limit of n-groupoids, as the natural framework for such analysis.22 Grothendieck is here demonstrating the elasticity of categorical notions and taking full advantage of the reflexivity of categories. He has been somewhat critical of the mathematical community in general for not pursuing this type of idea in spite of the foundations laid for the effort by the work of his students L.Illusie and J.Giraud as well as

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Quillen’s original work.23 In contrast, the tenor of the times is perhaps better reflected in the work on what might be termed the homotopy theory of tensor categories in which both physicists and mathematicians have participated over the past decade.24 The generality of Grothendieck’s approach and his penchant for introducing new categorical constructions are somewhat at odds with contemporary focus away from the abstract and the general and towards the particular and the concrete. It is easy to forget, in the current drive for concreteness, that Grothendieck’s ‘abstractions’ were not introduced for their own sakes but rather to render mathematical arguments more flexible and to avoid choosing in any arbitrary way a fixed starting point. 6.6 Generality and mathematical wholes The power and beauty of ideas such as Weil’s varieties, Grothendieck’s schemes, Arakelov theory and homotopical algebra and the strength of the results to which they have led make it very difficult to propose criticisms or improvements. But the difference in approach to generality between Euclid on the one hand and Weil and Grothendieck on the other may give pause for reflection. Both Weil and Grothendieck seek to include in their theories a broader range of objects than had previously been possible (naturally without any sacrifice in rigour). The differences in attitude that have been noted between Weil’s intention to create a theory which remained as close as possible to those which he had inherited and Grothendieck’s reconstruction of algebraic geometry on categorical footings merely underline the similarity of these two enterprises when compared with that of Books XI–XIII of the Elements. It is perhaps too readily accepted that a generalization of one theory by another in mathematics must involve the introduction of new classes of objects which include (as special cases) the objects of the previous theory. Hence Weil’s varieties generalized classical algebraic geometry by (among other things) permitting geometric objects to be considered over fields of positive characteristic as well as over the complex numbers, while Grothendieck’s schemes include Weil’s varieties as ‘integral reduced schemes of finite type over a field’. At times the introduction of new objects can serve a useful purpose in isolating and axiomatizing certain properties which had previously been observed only in combinations which may have obscured their essential nature. However, such strategies generally have more to do with introducing or developing new subject matters than finding a sense of completion in a given theory. In the interpretation given here, the principal objective of the reformulations of algebraic geometry discussed above

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was less the discovery of a new subject matter than the completion of an existing one so as to permit investigations which had already commenced to find a natural conclusion. In such a context there may be a role for a notion of generalization which does not involve solely the introduction of a broader range of objects but in which the new objects serve as a framework in which things already defined and studied can be put together to form new wholes. In other words, a sense of generalization based less on the increased ‘generality’ of the new objects introduced than on their character as complete wholes. In many ways both Weil and Grothendieck (and the mathematicians who have subsequently developed the theories discussed above) have shown themselves sensitive to this kind of approach. For example, as has already been indicated, the objects in Arakelov theory, the natural outgrowth of Weil’s musings on the analogies between number fields, function fields and differential geometry, are composed of more classical objects as parts. What is lacking to date is a notion of the wholes which they constitute. All that is suggested here is a change in the view of what a successful outcome might entail. Perhaps there is an approach to these complex objects which does not completely absorb the previous theories, as Weil indicates, but rather incorporates them as Books I–X are incorporated in the objects considered in Books XI– XIII in the Elements. Similarly, Grothendieck too sees the internal structure of the categories involved in his Postnikov tower, that is to say their parts, as of the same level of importance as the properties of the wholes axiomatized by Quillen’s homotopical algebra. In his case the problem is in some sense opposite to Weil’s in that it is not so much that the properties of the parts are absorbed in the new whole as that any fixed notion of whole is viewed with great suspicion. Again the matter at issue has largely to do with the notion of what a successful outcome of the theory might be. In Grothendieck’s case this would appear to be not so much a new theory absorbing the previous ones, as Weil suggests, but a wider system of relationships between the objects. This insistence on continual relativization, however, may arise more as a reaction to a too fixed view of the nature of wholes and parts taken in certain mathematical quarters rather than stemming from problems intrinsic to this relation. Euclid’s wholes—solid figures—are neutral and are not to be taken as establishing ‘once and for all’ the destinies of their parts or somehow imposing arbitrarily and from without a fixed set of characteristics. Would the inclusion of this sort of whole, establishing natural contexts in which distinctions of coefficient and variable, base and extension could be made relative to the given context,25 although contrary to Grothendieck’s continuous relativization, in fact vitiate the whole enterprise?

Conclusion

The objective of this investigation has been to demonstrate how mathematical texts may be read as texts and to examine some of the strategies of argumentation that become visible on such readings. In pursuing this objective I have not made use of the ideas and methods developed by practitioners of modern hermeneutics although it would not be difficult to point out certain commonalities of interest and outcome.1 The introduction of grand theory into the investigation at this time would merely distract attention from what should be the (rightful) centre of attention, the mathematical texts themselves. Mathematical understanding is achieved by finding appropriate mathematical terms in which problems can be formulated and theorems enunciated. It might be said that the mathematics in a mathematical text is precisely that in it which can be translated into other (mathematical) terms.2 The text and its argument is what cannot be translated or so transformed. These fixed textual elements, figures of thought, have been the object of study in this book. Mathematical texts are both difficult and interesting to read as texts because of the great disparity between their mathematical and textual aspects. It is possible to extract from what has gone before certain themes which group these textual elements together and such themes may be useful in summarizing what has been said and in pointing out directions for future investigations. The first theme has to do with mathematical subject matters. The philosophy of mathematics has occupied itself with the nature of mathematical objects and our knowledge of them, but has by and large taken little notice of the nature of mathematical subject matters. It is evident to the most casual historical investigator that these have altered over the years, sometimes dramatically and rapidly, at other times slowly and calmly. It is tempting to link these evolutions to mathematical discoveries themselves but this would not give an adequate account of developments. Consider, for example, Hilbert’s Zahlbericht in 1894 which dramatically altered the subject matter of number theory without introducing fundamentally new facts. Of course

158 CONCLUSION

not every attempt to state or present a mathematical subject matter has been successful in convincing other mathematicians to continue the investigation. Kronecker’s Grundzüge is but one instance of such failures. The student of mathematical subject matter therefore cannot rely on the accepted notions of what, say, number theory ‘is about’ but must explore each text to see how it goes about organizing its material. The subject matter of the Elements is discovered and structured in Definitions, Postulates and Common Notions and is expressed in Propositions which are both theorems and problems. This subject matter is the result of a sustained investigation into the nature and possibility of definition and the role of limits and boundaries in producing definable things and their definitions. The price to be paid for the achievement of a subject matter which is in this way self-sufficient is that demonstrations must be conducted in the same terms as the definitions and therefore cannot address concerns which go beyond these terms. Descartes and Hilbert seek subject matters which can address the issues which Euclid is forced to leave to one side. These subject matters are the result of investigation into the nature and possibility of mathematical construction and demonstration and the role of the mathematician in generating and characterizing mathematical objects. The price to be paid for the achievement of subject matters based on methodical constructors or axiomatized theories is a set of paradoxes and conundra regarding the existence of mathematical objects and their nature. Such questions have been the staple of philosophy of mathematics for many years—need it be remarked that consensus has not been achieved? Have these questions acquired a privileged position because they are somehow inherent in mathematics itself or because we have failed to reflect on the possibility of mathematical subject matters which might be differently organized? The second theme which emerges from this investigation is that of the development of mathematical methods. This does not refer, strictly speaking, to mathematical techniques but rather the ways in which subject matters are developed and extended. Euclid’s method is one of multiple contexts or points of view. At each stage in the Elements material which has already been studied is placed in new contexts which enable additional and usually more precise determinations to take place. In discussing this aspect of Euclid’s work I have focused on Books V–X, but the same approach is evident throughout the treatise. The fine discriminations of Book X are the result not of the introduction of new entities or new materials but the examination of ‘old’ materials in new contexts. There is a place in this method for both continuous and discrete, rational and irrational. The price to be paid is that

CONCLUSION 159

totalities of objects such as ‘the integers’ or ‘plane figures’ escape examination. Dedekind and Kronecker sought to explain and expand the work of predecessors such as Gauss, Dirichlet and Kummer. To accomplish this they deliberately set aside older materials and constructed or reconstructed mathematics from the beginning. Dedekind did so in the belief that the human spirit was ‘of a god-like sort’3 in that it contains a kind of ‘creative power’4 and that mathematics, as a part of logic, was a free creation of this spirit. Dedekind’s mathematics is a spinning out from the mind itself of mathematical concepts from sets to integers to rationals to real and complex numbers, each of which can be viewed as another creation of the human spirit. This is a homogeneous sequence which can be followed in either direction. Kronecker, in direct contrast, saw mathematics as, in the first instance, a creation of God which humankind (another of God’s creations) could and should make use of. To utilize this marvellous tool is to calculate and it is the objective of the mathematician to increase the available powers of calculation. Kronecker’s mathematics is a development of the means of calculation beginning with the irreducible minimum, the natural numbers and, like Dedekind, moving to increasing levels of complexity but, unlike Dedekind, not to new systems or entities, just to more powerful computational means. As a result Kronecker’s sequence cannot be followed in both directions, it leads inexorably towards higher levels of complexity but provides no way to go ‘behind’ the natural numbers. The price to be paid for these homogeneous systems reconstructing mathematics from the ‘ground up’ as it were, is that the rational and the irrational, the discrete and the continuous are placed in opposition and the mathematician must choose which is ‘prior’ and which is explained in terms of the other. Gödel’s destruction of Hilbert’s attempt to marry Dedekind’s conceptualism with Kronecker’s finite computability in no way acted as a brake on the ‘foundationalist’ enterprise although the urgency felt has long since drained away. Is there necessarily a single ‘homogeneous’ development of mathematics (whether bi-directional as Dedekind’s or uni-directional as Kronecker’s)? Or is it possible that Euclid’s plural contexts and multiple points of view provide a more accurate reflection of mathematicians’ procedures or at least a more useful model to follow? The final theme is that of mathematical systems involving the sense of conclusion or closure in a mathematical investigation. In Books XI– XIII Euclid views his measured plane figures as parts of solid figures which act as wholes simply because they in turn are not parts of something else. His investigation achieves closure by examining these wholes and showing that they come in a limited number of varieties which can be studied and compared both with a common measure and

160 CONCLUSION

with each other. The previous enquiries are not subsumed or displaced by this new one but rather find a natural place in providing measured parts. As the study of plane figures and their measures has no completeness in itself, it remains available for other enquiries and retains its integrity ‘in spite of’ becoming part of the study of solid figures. The price for this type of conclusion is the limitation to specific types of figures. Weil and Grothendieck set about to expand the range of objects and circumstances that algebraic geometry can rigorously examine. They wished to generalize previous theories to cover a variety of instances which were previously either ignored or dealt with only by analogy to known cases. Weil’s wholes are universal containers which subsume within themselves all of the objects of study. They do so not merely by inclusion but by a process of inclusion and distinction which permits the introduction of new objects which are built out of known materials. This process removes distinctions previously thought apparent and brings all of the objects in question under a single, universal theory. Grothendieck’s wholes are relative in that each is self-sufficient and can be related to all others in ways which demonstrate reflexivity. No universal distinction between variable and coefficient or subject and predicate can be sustained and wholes can be recognized precisely because they exhibit this self-sufficiency. Mathematical theories achieve completeness by establishing the widest freedom of reference between objects and by consideration of the broadest domains in which the questions to be studied can be set. Grothendieck’s generalizations, like Weil’s, incorporate previous theories by exhibiting them as sub-categories satisfying certain special conditions. General or philosophical discussions of ‘abstractness’ in mathematics are not frequently carried out at a high level and most mathematicians, accustomed to the use of abstractions and aware of their benefits, do not find such discussions as do occur very compelling. Yet there is no doubt that contemporary mathematics is going through a phase of reaction against what is perceived as ‘abstract mathematics of the 1960s’ and that the consequences of this reaction, at various depths, are far from clear. Is generalization and abstraction the only way to completeness and closure in mathematics? Are previous theories bound to be subsumed and ‘swallowed up’ by later ‘more general’ or ‘more abstract’ versions? Or can we find a place for objects which are wholes and which incorporate other objects as parts without destroying or altering their identities? These themes and questions emerge from the rather limited exploration of the Elements and other mathematical texts which has been undertaken here. There is little doubt that many more themes and

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questions—perhaps even a few answers—can be found in more extensive studies of mathematical texts.

162

Notes

1 The opening of the Elements 1 The principal textual references for the Elements will be Euclidis, Elementa, post I.L.Heiberg, ed. E.S.Stamatis (Leipzig 1970) in five volumes including Scholia (referred to as Heiberg); and in English, The Thirteen Books of Euclid’s Elements, trans. T.L.Heath (2nd edn, Cambridge 1925; reprinted, Dover 1956) (referred to as Heath). The more recent Euclide, Les Elements, vol. 1, trans. M.Caveing and B.Vitrac (Paris 1990) (referred to as Vitrac) brings Heath up to date. All translations are from Heath unless otherwise noted. 2 It should be noted that the numbering of definitions is not found in the best manuscripts of the Elements although there is some grouping of the definitions. 3 The usual Greek term for ‘principle’ is which also means ‘beginning’. Cf. Heath’s brief discussion of first principles in Greek science, Heath, vol. 1 pp. 117–24. 4 Although these remarks may remind some readers of the ‘formal’ definitions used by Hilbert and logicians who have followed him, there are significant differences between Euclid’s and Hilbert’s approaches as will be discussed below. Philosophers of science and logicians seem to have forgotten some of the other types of ‘formal’ procedures with ‘empty terms’ that have been employed in other disciplines since ancient times. General discussions of such ‘topics’ can be found in R.McKeon, ‘Creativity and the Commonplace’, in M.Backman (ed.) Rhetoric: Essays in Discovery and Invention (Ox Bow Press 1987), and R.McKeon, ‘Arts of Invention and Arts of Memory: Creation and Criticism’, Critical Inquiry, 1, (1975): 723–41. The formal process that Euclid follows is structured around the topics of whole and part, measure and measured, container and contained. The general question of the topical organization of mathematical arguments will not be pursued here. 5 See, for example, the discussion of ‘Pure Being’ at the opening of G.W.F. Hegel, Science of Logic, trans. A.V.Miller (London 1969). The philosophical ramifications of Euclid’s method will not be pursued here.

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6 The English word carries so many connotations and meanings that are at variance with Euclid’s usage that it seems almost certain to mislead the reader. On the other hand use of any other word to translate this basic term would seem deliberately obscurantist. The reader is asked to apply the argument in the text as rigorously and strictly as possible and to place in suspense as far as possible the semantic connections the word ‘length’ may carry. 7 The Greek is meaning extremities or limits. 8 As Heath and others suggest there were many definitions of ‘straight line’ available at the time, based on very different approaches to the establishment of a subject matter for geometry. It is not too much to say that one could classify these various approaches through analysis of the definition of straight line. 9 This point of view is too well represented in the history of commentary on the Elements to require detailed citation. 10 Vitrac states flatly that ‘the three definitions for surfaces are rigorously parallel to those for lines’, Vitrac, p. 156. 11 Heath, Vol. 1, p. 176. 12 See the discussion in Part III of the definitions of angle in Book XI. 13 This is one in what is a long series of instances in which Euclid avoids an entitizing definition in favour of one which establishes new relationships between existing things. The word ‘context’ will be used (at the risk perhaps of overuse) to describe most of these definitional moves. 14 Far from having a univocal meaning the term ‘part’ will be shown to have an extraordinary series of meanings in the Elements. The role of ambiguity in mathematical argument has been dismissed so thoroughly by philosophers that it may be useful to begin a re-examination. If so, Euclid’s use of ‘part’ should be one of the first case studies. 15 Cf.Aristotle, Categories 11b15–13b35 (the so-called ‘Postpredicaments’). Aristotle is also on record as stating that it would be absurd to try to define ‘figure’ in general over and above the particular figures, De Anima III, ch. 1. For him ‘figure’ and ‘soul’ seem to have much in common. 16 The standard edition is I.L.Heiberg and H.Menge (eds) Euclidis Opera Omnia, vol. VI (Leipzig 1896). A recent translation of a mediaeval Latin version can be found in S.Ito, The Mediaeval Translation of the Data of Euclid (Berlin 1980). Given the very different orientations of the two books it is dangerous to rely on one as a guide to the interpretation of the other. A general study of Euclid’s thought, including the remaining books on optics as well as the mathematical texts, would be very welcome. 17 In particular the discussion of whether a boundary must be ‘closed’ or not is of little import (e.g. Heath, vol. 1, pp. 182–3 and Vitrac pp. 160–1). It is perfectly plausible to consider the things defined in Definiitions 1–12 as ‘figures’. The point is that they can be defined without recourse to the general concepts of boundary and figure, this is not the case with the things defined in Definitions 15–23. 18 Once again, a discussion of the phrase ‘in one plane’ is postponed until Part III.

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19 The Greek for Common Notions has usually been interpreted with notions from Stoic philosophy in mind. In particular the ‘commonality’ in question seems to be most frequently interpreted as ‘common to all mankind’. The Common Notions are then thought to be a kind of substitute for ‘axioms’ (a term common to Stoic and Aristotelian traditions). The interpretation provided here is compatible with a range of views on Euclid’s relationship with various philosophical schools of antiquity and suggests connections with the mediaeval logic of ‘syncategoremata’. 20 One could pursue this analogy by aligning the definitions with substantives and the propositions with verbs. Current analysis of the grammatical ‘parts of mathematical speech’ seems to be limited to the predicates of first-order logic and certain analyses of ‘types’.

2 Propositions and proofs—theorems and problems 1 Heath, vol. 1, pp. 124–9, Vitrac, pp. 133–7 and A.C Bowen, ‘Menaechmus versus the Platonists: Two Theories of Science in the Early Academy’, Ancient Philosophy, 3, (1983): 12–29. 2 Proclus, A Commentary on the First Book of Euclid’s Elements, trans. G. Morrow, (2nd edn, Princeton 1992), pp. 63 et seq. 3 The literature on this text is far too extensive 10 be summarized here. D. R.Lachterman, The Ethics of Geometry , (Philadelphia 1989), provides a thorough review of Descartes’ place in the development of modern mathematics and philosophy. The discussion presented here also owes much to H.J.M.Bos, ‘On the Representation of Curves in Descartes’ Géométrie’, Archive for the History of Exact Sciences 25 (1982): pp. 295– 338; A.G.Molland, ‘Shifting the Foundations: Descartes’ Transformation of Ancient Geometry’, Historia Mathematica, 3, (1976): 21–49; and T. Lenoir, ‘Descartes and the Geometrization of Thought: The Methodological Background to the Géométrie’, Historia Mathematica, 6, (1979): 335–72. 4 Descartes, Géométrie, p. 3. All references will be to The Geometry of René Descartes, trans. D.E.Smith and M.L.Latham (Dover 1954) (to be referred to as Géométrie) which includes a facsimile of the first (French) edition. All translations are mine. 5 ibid. p. 3. 6 Descartes has no use for Euclid’s carefully structured distinctions between line, surface and solid. Modern dimensional analysis stems from this reduction of quantity to a single type. 7 ibid. p. 8. 8 ibid. p. 11. 9 Or perhaps by metaphor? The manner in which the geometer ‘reads’ figures may be literal or analogical. There are interesting ‘parallels’ with literary theory here and connections with Descartes’ narrative method in the Meditations and elsewhere.

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10 Or perhaps an equation. The manner in which the sign ‘=’ functions in algebra may be seen ‘literally’ or ‘figuratively’. See note 9. See also S. Buchanan, Poetry and Mathematics (Chicago 1962). 11 ibid. p. 43. 12 See especially Bos, ‘On the Representation of Curves’. 13 Géométrie, p. 49. 14 ibid. pp. 54–7. 15 ibid. p. 49. 16 ibid. p. 90. 17 ibid. p. 92. 18 ibid. pp. 152–5. 19 ibid. p. 241. 20 All references will be to D.Hilbert, Grundlagen der Geometrie, translated as Foundations of Geometry, trans. L.Unger (Open Court 1971) (referred to as GdG). GdG has received much less critical and philosophical attention than Hilbert’s later logical work. A useful survey of the background to the book can be found in H.Freudenthal, ‘Zur Geschichte der Grundlagen der Geometrie’, Nieuw Archiv. voor Wiskunde, 4, (1957): 105–42, and the reception of the book is discussed in detail in G. Birkhoff and M.K.Bennett, ‘Hilbert’s Grundlagen der Geometrie’, Rendiconti del Circolo Matematica di Palermo Series II, XXXVI (1987): 343–89. 21 GdG, p.2. 22 ibid. p. 2. 23 A translation of this speech is provided in the American Mathematical Society (AMS) Hilbert Symposium Volume, F.Brovder (ed.), Contemporary Mathematics Related to Hilbert’s Problems, Proceedings of the Symposia in Pure Mathematics XXVIII, (New York 1976). 24 D.Hilbert and E.Bernays, Grundlagen der Mathematik I, 2nd edn (Berlin 1938), pp. 2–3 (Translation mine). 25 D.Hilbert, ‘Axiomatisches Denken’ in Gesammelte Werke, vol. 3, (New York 1965). 26 That is, if Desargues’ theorem does not hold then a plane geometry satisfying Hilbert’s axioms cannot be embedded in a space geometry satisfying the same axioms. 27 This is equivalent to the axiom of commutativity for the algebraic structure which ‘models’ the geometry being studied. 28 GdG, pp. 26–8. 29 The question of what this study does add up will be addressed in Part III. 30 31 Such as the well-known objection to the demonstration of Proposition 1 discussed below, that Euclid nowhere proves that the circles do in fact intersect. 32 Is mathematics entirely exempt from the difficulties surrounding the ‘quest for certainty’ adumbrated by twentieth-century philosophers from Dewey to Kolakowski? 33 See, for example, Heath, Vol. 1, p. 249.

NOTES 167

34 A similar view of the role of Proposition 47 in the argument of Book I of the Elements is given in remark at Section 256 of G.W.F.Hegel, Philosophy of Nature trans. Miller and Finlay (Oxford 1970) pp. 31–3. 35 Any attempt to provide a thoroughgoing analysis of the Elements would have to follow these lines of development quite carefully.

3 The contexts of measurement 1 I do not here wish to be understood as making any statement as to the use or lack of use that Euclid may have made of the work of his predecessors. Euclid’s enterprise is that of founding a science; this has nothing to do with his place in a historical sequence. 2 The ambiguity in the use of the term is crucial. In particular it would be inappropriate to refer to these contexts as undefined ‘primitives’ because of these shifts in meaning. The classical rhetorical doctrine of ‘topoi’ or places provides the best background for this notion. 3 Of course one might also relate figures to each other in terms of equals/ greater than/less than, but such comparisons are not based on comparing, say, hexagons to hexagons, but rather cut across types of figures. Circles play a special role in these comparisons. 4 Book V of the Elements has been commented upon more frequently in recent times than any other portion of the text. W.R.Knorr, The Evolution of the Euclidean Elements (Norwell 1975), I.Mueller, Philosophy of Mathematics and Deductive Structure in Euclid’s Elements (Massachusetts 1981), D.R.Lachterman, The Ethics of Geometry (Philadelphia 1989), and F.Beckmann, ‘Neue Gesichstpunkte zum 5. Buch Euklids’, Archive History of Exact Sciences, 4, (1967): pp 1–144, give an idea of the range of investigations, historical, philosophical, philological and mathematical, that have been undertaken. The pattern of interpretation pursued here deviates consistently from these authors so that continuous reference to the literature would be more distracting than useful to the non-specialist reader. 5 But see Aristotle, Metaphysics 1023 b 12, cited in Heath, vol. II, p. 115. 6 The topic of space and place, although well used by ancient and mediaeval philosophers and scientists, has not fared well since the nineteenth century. Much of the discussion here centres on concepts of measurement and number and unjustly leaves space and place to one side. A detailed history of this topic with its rhetorical as well as scientific character could be of great value to mathematics. 7 This definition has not been well received by commentators. Heath’s discussion of Hankel’s comments are revealing in that many mathematicians would find no point in what they would likely term a vague definition of this sort. But then most mathematicians would have a preconceived notion of what a ratio is in the first place. 8 See, for example, Hilbert, GdG, pp. 41–3. 9

168 NOTES

10 Relationships of greater than/less than/equal to but expanded to encompass the magnitudes which measure and are measured by the magnitudes ‘in the ratio’. 11 The failure to recognize Euclidean ratios and proportions as relationships between measured things rather than as entities is as widespread among commentators on Book V as the failure to identify the role of figures in Book I. It would be interesting to compare Euclid’s approach here with Aristotle’s use of ‘figures’ and ‘terms’ of syllogisms in the Prior Analytics. 12 Here the eighteenth-century English commentator Robert Simson has noticed the difference in treatment between ratios and the measured things of Book I and criticized Euclid for not including ‘axioms’ which relate the behaviour of multiples to their parts (measured things to that which measures them) in relation to greater than/less than/equal to. Simson’s suggested ‘axioms’ amount to elaborations of the Common Notions and, although they are implicit in the notion of ‘equality’ sketched above, their inclusion might indeed render Euclid’s argument more transparent at a number of points in Book V. The risk in doing so, however, would be to make ‘magnitude’ into an entity in itself, a temptation to which Simson seems to have succumbed from time to time. 13 See Heath, vol. 2, p. 170. 14 See Mueller, Philosophy of Mathematics, ch. 3 for a useful summary and analysis of Book V as an attempt to ‘axiomatize’ proportionality and a discussion of the extent to which modern notions of ‘constructibility’ can account for Euclid’s argument. As the thesis presented here is that magnitudes, ratios and proportions are not things but relationships among things, the details of Mueller’s analysis (as well as Beckman’s very thorough treatment, in ‘Neue Gesichstpunkte zum 5’) need not detain us. 15 A more detailed study of the sequence of propositions in Book V following this outline is desirable but would lead too far from the line of argument being pursued. 16 See Mueller, Philosophy of Mathematics, ch. 2, for a useful summary of this obscurity. 17 Hence Books VII–IX may not be as ‘isolated’ in the Elements as Mueller suggests. Ibid. p. 58. 18 See, for example, Propositions 18 and 19 of Book IX where the grammar of the Propositions takes the form ‘investigate whether it is possible to find ’
so that they remain nominally within the QED framework. 19 See Heath, vol. 2, pp. 341–344, and Mueller, Philosophy of Mathematics, pp. 74–82 (passim). 20 Largely late and unreliable. 21 ‘the underlying’ or ‘the laid down’, is Euclid’s standard term for ‘hypothesis’ and recurs in Book XI. A full study of the semantics of (not only in Euclid but in this word and its relationship to Aristotle and other Greek philosophers) would be welcome. 22 It should be clear that the interpretation provided here differs from some recent readings in locating the relationship between Book X and the

NOTES 169

preceeding books in the interplay of various contexts of comparison of geometric material. This discussion, however, by no means constitutes a complete analysis of this book which is well known as one of the most difficult in the Elements. Some of the recent debate on this part of the text can be sampled in Mueller, Philosophy of Mathematics, pp. 107–13, D.H. Fowler, ‘Investigating Euclid’s Elements’, British Journal for the Philosophy of Science, 34, (1983): pp. 57–70; D.H.Fowler, The Mathematics of Plato’s Academy (Oxford 1987), pp. 166–90; and D.H. Fowler, ‘An Invitation to Read Euclid’s Elements Book X’, Historia Mathematica, 19, (1992): 221–42. 23 Heath, vol 3, p. 15. 24 Ibid. p. 24.

4 Number theory in the nineteenth century 1 The collection Numbers, ed. H.-D.Ebbinghaus, H.Hermes, F. Hirzebruch, M.Knecher, K.Mainzer, J.Neukirch, A.Prestel and R. Remmert (New York 1990), provides a useful overview and historical remarks on a wide range of mathematical issues related to the story recounted here. See also H. Stein, ‘Logos, Logic and Logistiké: Some Philosophical remarks on the 19th Century Transformation of Mathematics’, in D.Rowe and E.J. McCleary (eds) History of Modern Mathematics Vol. I: Ideas and Their Reception (London 1989), pp. 238–59. 2 Lachterman, The Ethics of Geometry, refers to these questions but a full treatment would have required a second volume of his book which he did not live to complete. 3 See L.Kronecker, ‘Uber den Zahlbegriff’, in L.Kronecker (ed.) Mathematische Werke, vol. III, band 2, (New York 1968), p. 253. My translation. 4 See Stein, ‘Logos, Logic and Logistiké’, for further remarks in this connection. A recounting of the history of nineteenth-century mathematics in terms of the movement towds ‘arithmetization’ in this broad sense could be one of the more important and useful undertakings available to historians of mathematics. 5 For a simple proof see L.Washington, Cyclotomic Fields (New York 1982), p. 22. For the circumstances of Kummer’s discovery see, for example, P. Ribenboim, The Work of Kummer on Fermat’s Last Theorem’, in N. Koblitz (ed.) Number Theory Related to Fermat’s Last Theorem (Berlin 1980). 6 See the overview of this work given by A.Weil in his introduction to E.E. Kummer, Collected Papers (New York 1976). 7 The collected papers are in P.G.Lejeune-Dirichlet, Mathematische Werke (New York 1969), but Dirichlet’s work as a whole remains woefully underexplored. The various editions of Dedekind’s reworking of Dirichlet’s lectures on number theory will be discussed below.

170 NOTES

8 See, for example, Weil, Collected Papers, p. 10. Also, N.Bourbaki, Commutative Algebra Chapters 1–7, (New York 1989), p. 584. 9 See especially, H.Edwards, ‘The Genesis of Ideal Theory’, Archive for History of Exact Sciences, 23, (1980): 321–78 (hereinafter Edwards, ‘Genesis’); H.Edwards, ‘Dedekind’s Invention of Ideals’ Bulletin of the London Mathematics Society, 15, (1983): 8–17; and H.Edwards, O. Neumann and W.Purkert, ‘Dedekinds Bunte Bemerkungen zu Kroneckers Grundzüge’, Archive for History of Exact Sciences, 25, (1981): 49–85 (hereinafter Edwards et al., ‘Bunte Bemerkungen’). 10 “
von einem einfachen und zugleich strengen und völlig allgemeinen Gesichtspunkt
’ R.Dedekind and W.Weber, ‘Theorie der algebraischen Funktionen einer Verändlicher’, in R.Dedekind (ed.) Mathematische Werke vol. 1, ed. E.Noether, O.Ore and R.Fricke (New York 1969), pp. 238–350 (referred to hereinafter as Dedekind, Werke). 11 Dedekind, Werke, vol. 3, pp. 315–35. See note 12. 12 ibid. pp. 297–315. Both Stetigkeit und irrationalen Zahlen and Was sind un was sollen die Zahlen are available in Englsh translations in R. Dedekind, Essays on the Theory of Numbers, trans. W.W.Deman (Mineola 1958). 13 ‘Sur la Théorie des nombres entiers algébriques’, ibid. pp. 262–96 (see especially p. 269). 14 ‘Uber die einführung neuer Funktionen in der Mathematik’, ibid. pp. 428– 39. 15 See R.Dedekind, ‘Eine Vorlesung über Algebra’ in W.Scharlau (ed.) Richard Dedekind 1831–1981 (Wiesbaden 1981), pp. 59–101. 16 The English translation ‘field’ is far from the sense of ‘body’ or ‘organism’ that is carried by the German and that Dedekind had in mind. 17 R.Dedekind (ed.) Vorlesungen über Zahlentheorie von P.G.Lejeune Dirichlet, 4th edn (New York 1968), 11th supplement, section 160, p. 452, hereinafter Dirichlet—Dedekind (edition numbers will be specified where necessary). 18 The later sections of the supplement, dealing with topics such as the class number formula, remained virtually unchanged throughout this process. While these topics are generally considered today as being of a more advanced nature than the basic algebraic structure of number fields, they were in fact dealt with by Dirichlet and Dedekind’s treatment merely translates his predecessor’s work in a new formulation and language. Dedekind’s most elaborate editorial efforts went into explaining the basics of fields, the notions of ‘conjugate’, ‘module’, ‘ideal’, ‘norm’ and ‘discriminant’. 19 This is Edwards’s principal focus. 20 Another useful source of historical information is W.J.Ellison and F. Ellison, Théorie des nombres’ (especially Section V, ‘La théorie des nombres algébriques’) in J.Dieudonné (ed.) Abrégé d’Histoire de Mathématiques 1700–1900, (Markgröningen 1978). A more modern treatment which none the less retains links to Dedekind’s era can be found in H.Hasse, Number Theory (New York 1978). 21 Dedekind, Werke, vol. 1, pp. 40–68.

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22 Technically Dedekind considers a polynomial ring over Q modulo a double modulus consisting of a polynomial and a prime. For a discussion of the difficulties encountered by other mathematicians attempting similar but less careful analyses see N.Bourbaki, Commutative Algebra: Chapters 1–7 (New York 1989), p. 586. 23 This is in fact the ring of algebraic integers in the field of numbers a+bβ where a, and b are rational numbers. 24 That is, as an ideal in the ring of algebraic integers of the field. Of course at this stage neither Dedekind nor Kummer before him had yet developed the notion of algebraic integer hence neither had a notion of ‘ring of algebraic integers’ within a ‘field of algebraic numbers’. This makes no difference in the case of cyclotomic numbers (see the appendix to this chapter). Kummer left the notion of ideal factorization as a computational procedure without providing any further enlightenment or definition. Dedekind’s early work follows Kummer’s approach to these matters fairly faithfully. 25 The standard modern treatment of this material can be found in Washington, Cyclotomic Fields. 26 See below, p. 102 27 Dedekind, Werke, vol. 3, pp. 399–407 (especially pp. 404–7) 28 E.Kummer, ‘Uber die allgemeinen Reciprocitätsgesetze unter den resten und nichtresten der Potennzen, deren grad eine Primzahl ist’ in E.E. Kummer, Collected Papers, (New York 1976), pp. 699–840 (see especially p. 735 where he wonders whether ‘primes which divide the discriminant’ deserve the name ‘prime’). 29 See the notice to the second edition of Dirichlet—Dedekind (1871). 30 There is possibility for confusion here since the technical definition of ramification has not been given (see the appendix to this chapter). As noted below in the discussion of Kronecker’s work, in a technical sense all number fields which are extensions of the rational numbers are ‘ramified’. However, as further noted below, the ramification exhibited by Kummer’s cyclotomic numbers is quite ‘benign’ compared with that found by Dedekind in more general number fields. Hence it may be justifiable to refer to Dedekind’s work in terms of the ‘discovery’ of phenomena of ramification. 31 Dedekind, Werke, vol. 3, pp. 313–14. 32 See, for example, ibid. p. 225. A ‘finite field’ (we would say number field) is here defined as one with a finite number of subfields. Ibid. p. 224. 33 ibid. p. 236. 34 Of course, as noted above, they had dealt with fields where, in effect, nothing was lost in ignoring the ‘fractional’ part (i.e. there was nothing to be gained by introducing the integral/fractional distinction). 35 ibid. p. 251 (translation mine). 36 Much of his article, ‘The Genesis of Ideal Theory’, is taken up with these matters. Edwards’s question is quite pointed in that Dedekind’s subsequent work does not provide many new number theoretic facts. My intention in recounting the story is to show how Dedekind’s ideas

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37 38 39 40 41 42

43 44

45 46

47 48 49 50

51 52 53 54 55 56 57 58 59 60

concerning the nature of number fields developed and how this altered his view of the mathematical facts he had in his hands as early as 1871. ‘Sur la théorie des nombre entiers algébriques’, Werke, vol. 3, pp. 262–97. ibid. p. 268. ibid. p. 469. ‘Uber den Zusammenhang zwischen der theorie der Ideale und der Theorie der höheren Kongruenzen’, ibid. vol 1, pp. 202–33. ibid. p. 203 (translation mine). ibid. vol. 3, p. 271. This discussion has reference to so-called ‘fractional ideals’ or, in modern terminology, projective rank one modules over the ring of integers. ibid. vol. 1, pp. 351–97. This is a simplification. Dedekind’s treatment is not restricted to the absolute case but covers the relative case of the extension of one number field by another. In this general set-up the discriminant is in fact an ideal. But Dedekind’s theorem on the factorization of ideals ensures a similar finiteness result. He is unable to treat so-called ‘wild’ ramification. Although it is the end of the story told here, this was not the end of Dedekind’s thinking about his ideal theory. Edwards has a fascinating analysis of a later paper of Dedekind’s which might have been used to provde a more ‘Kroneckerian’ view of the subject. See Edwards, ‘Genesis’ especially Section 13. See the third-hand version in Dedekind, Werke, vol 3, p. 481. All translations mine. L.Kronecker, Mathematische Werke, vol. 2, (New York 1968), pp 237–389 (referred to hereinafter as Kronecker, Werke). This is discussed in his ‘Genesis’ noted above and developed extensively in his book Divisors (Berlin 1990). Edwards discusses these matters briefly in his ‘Kronecker—An Appreciation’, Mathematical Intelligencer, 9, (1987): 35–47. The other source for this side of Kronecker’s work is A.Weil, Elliptic Functions According to Eisenstein and Kronecker (New York 1975), which is required reading for students of Kronecker, nineteenth-century mathematics, contemporary number theory, etc. L.Kronecker, Vorlesungen über Zahlentheorie I, ed. Hensel (New York 1978). A.Weil, Oeuvres scientifiques, vol 1. (New York 1980) , pp 442–54. Proceedings of Symposia in Pure Mathematics XXVIII (New York 1976). Kronecker, Werke, vol III, part 1, pp. 249–75. ibid. p. 272. ‘Uber einige Anwendungen der Modulsystem’, ibid. p. 156. See Edwards, Divisors, pp. 13–19. ‘Ein Fundamentalsatz der allgemeinen Arithmetik’, Kronecker, Werke, Vol III, part 1, p. 209. Edwards, Divisors, p. 27. Archive for History of Exact Sciences, 27, (1982): pp. 49–85.

NOTES 173

61 E.E.Kummer, ‘Uber die allgemeinen Reciprocitätsgesetze unter den resten und Nichtresten der Potenzen, deren Grad eine Primzahl ist’, in E.E. Kummer (ed.) Collected Papers, Vol. 1, (New York 1976). p. 737. 62 Kronecker, ‘Grundzüge’, Werke, vol. II, p. 323. Edwards has reconstructed a Kroneckerian theory of ramification in Divisors, pp. 54–9. 63 Except in so far as he deals with the properties of cyclotomic and quadratic fields in Dirichlet—Dedekind. 64 See H.Minkowski, Geometrie der Zahlen, (New York 1965). The reader may wonder whether this implies that the cyclotomic extensions over Q are ramified as well. Confusingly the answer is yes, but, as remarked previously, the cyclotomic extensions exhibit the simplest type of ramification behaviour. A rational prime either stays prime, splits into separate prime factors (as many as the degree of the extension) or becomes totally ramified, i.e. it becomes the power of a single prime (the power once again being the degree of the extension), and it is easy to compute which of these possibilities will obtain for a given rational prime p. This is a very benign form of ramification. 65 Kronecker dealt with these ‘fields’ as bereiche given by taking equivalence classes modulo the equation whose roots generate the field. The abelian Galois group was represented as a collection of rational functions permuting the roots of the equation. In abelian extensions one can assign certain elements in the Galois group of the extension (so-called Frobenius elements) to prime ideals. While not known in this generality by Kronecker et al., this principle was certainly applied in specific cases and made abelian extensions a natural class to consider. 66 This is the well-known Kronecker—Weber theorem. For a brief historical discussion see Edwards, ‘Kronecker—An Appreciation’. A modern proof (although fundamental, the theorem is by no means elementary) can be found in Washington, Cyclotomic Fields ch. 14. See also O.Neumann, ‘Two Proofs of the Kronecker—Weber Theorem “according to Kronecker and Weber”’, Journal für Reine und angewandte Mathematik, 323, (1981): 105–27. 67 These come in a range of types and may include a certain specified amount of ramification. The term is Hilbert’s. 68 The history of this theorem is also reviewed in Edwards, ‘Kronecker—An Appreciation’. The standard modern treatment is G.Shimura, The Arithmetic of Automorphic Functions’ (Princeton 1971), ch. 5, and is quite difficult. 69 Edwards has pointed out that factorization into primes was less important for Kronecker than the finding of greatest common divisors. It would perhaps be more correct (although less mathematically idiomatic) to substitute the latter phrase for the former in the above discussion. 70 It is worth emphasizing this point since the modern mathematical use of these types of theorems typically involves a ‘one way’ movement from the analytic to the algebraic. 71 Weil, Elliptic Functions, ch. 9.

174 NOTES

72 See H.Edwards, ‘Kronecker’s Theory of Generalized Arithmetic’, Jahresbuch der DMV, 94, (1992): 130–41. Edwards focuses on the algebraic side of Kronecker’s thought. 73 See A.Pillay and C.Steinhorn, ‘Definable Sets in Ordered Structures’, Bulletin of the American Mathematical Society (New Series), 11, (1984): 159–162. 74 K.Hensel, ‘Untersuchung der Fundamentalgleichung einer Gattung für eine reele Primzahl als Modul und Bestimmung der Theiler ihrer Discriminante’, Journal für Mathematik, 113, (1894): 61–83. 75 The study of Kronecker’s mathematics remains at an early stage. The above remarks can give only a small hint of the scope of his work but try to emphasize the consistency with which he pursued his mathematical objectives in a wide range of fields. The accusation levelled at Kronecker has always been that he attempted to act as some sort of ‘mathematical censor’, banning or barring results and methods from mathematics on ‘philosophical grounds’. The above reading suggests ways in which his efforts can be seen as having an inclusive aspect rather than being merely exclusive. 76 Kronecker, Werke, vol. III, part 1, p. 274. 77 Although the detailed proof for the general case never made it into Dirichlet—Dedekind. 78 Dedekind does not seem to have concerned himself with the problem of whether a finite set of generators could always be found. This was of course one of Hilbert’s first great theorems. 79 This is discussed in detail in H.Hasse, Number Theory (New York 1962) section 25(g). 80 Following the Kummer—Kronecker—Hensel line of thinking the modern treatment considers ramification a ‘local’ phenomenon to be analysed prime by prime over the various local completions of the number field.

Part III —Introduction 1 There is of course the further complication that both men are still alive at the time of this writing.

5 Types of wholes 1 Among the many discussions of this theorem the non-expert reader may find J.P.Serre, Around the Theorem of Mordell-Weil (Wiesbaden 1988) and J.Silverman, Arithmetic of Elliptic Curves (New York 1988), ch. 8, the most helpful.

NOTES 175

2 A group is said to be finitely generated if there exist a finite number of elements in the group whose (finite) powers and products give all of the elements of the group. 3 All references for Weil’s papers are to A.Weil, Oeuvres scientifiques (New York 1980) in three volumes including a valuable series of comments provided by Weil at the time when these papers were collected and edited. Weil’s writings are rich with historical insight and meditation. A. Weil, Number Theory: An Approach Through History, (Berlin 1984), and A. Weil, Elliptic Functions According to Kronecker and Eisenstein, (New York 1974), provide the reader with the unusual opportunity of viewing contemporary mathematics from a historical perspective. 4 These are mappings defined by quotients of polynomial functions. The field over which they are defined is the field from which the coefficients of the polynomials are taken. 5 A complex curve can be viewed as a two-dimensional surface over the real numbers. If it is projective then the surface in question will be compact and the topological characteristics of such a surface are governed by its orientability (the sphere is and the Mobius band is not) and its genus, more commonly thought of as the number of ‘holes’ (the sphere has none, the torus or anchor ring has one and so on). As the surfaces corresponding to complex curves are all orientable, they are distinguished by their genus alone. Since a curve is given by an algebraic equation it is desirable to have an algebraic interpretation of genus. This was already available through the theory of algebraic function fields of one variable developed by Dedekind-Weber and others in the nineteenth century. 6 For a modern introduction see Serre, Around the Theorem of Mordell— Weil, pp. 12–20. 7 The reader will also note the presence of remarks indicating Weil’s impatience with the algebraic tendencies of the day which emphasized Dedekind’s ideals over Kroneckerian methods. Weil was to become one of Kronecker’s champions, as noted above in Part II, although his emphasis on universal domains, discussed below, can hardly be characterized as Kroneckerian! 8 The theorem also covers curves of genus 1 with one point ‘at infinity’ removed and curves of genus 0 with three points ‘at infinity’ removed. By ‘genus’ here is meant the genus of the normalization of the projective curves corresponding to the given affine curve. C.L.Siegel, ‘Uber einige Anwendungen diophantischer Approximationen’, in Gesammelte Abhandlungen, vol. 1 (New York 1966). See also Serre, Around the Theorem of Mordell—Weil, ch. 7. 9 Most notably the conjecture of Birch and Swinnerton-Dyer which is discussed in J.Tate, ‘Arithmetic of Elliptic Curves’, Inventiones Mathematicae, 23, (1974): 179–206. 10 E.Artin, ‘Quadratische Körper im Gebiet der höheren Kongruenzen’, in E. Artin, Collected Papers (New York 1965). 11 That is, a field F in which px=0 for some positive integer p and all x in F. 12 ‘Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren’, in Artin, Collected Papers.

176 NOTES

13 Complete, hence projective, and smooth. 14 The Oeuvres scientifiques contain not only the official research announcements and the final papers (publication of which was interrupted by, among other things, the Second World War) but also letters from Weil to his sister Simone and to Artin outlining his discovery in both technical and non-technical terms. 15 See J.Dieudonné, A History of Algebraic Geometry, (Belmont 1985) (and references therein), for background on Severi and the Italians. 16 This is an algebraic variety of dimension equal to the genus of the curve which carries a group operation which in turn is algebraic (i.e. can be described by polynomial equations). A detailed expose of the development of the ideas involved in this construction can be found in C.L.Siegel, Topics in Complex Function Theory (New York 1969), particularly vols 1 and 2. Further details can be found in D.Mumford, Curves and Their Jacobians (Michigan 1975). 17 More specifically, the type of singular point illustrated in the text is known as a ‘cusp’ and may be contrasted with a nodal point where two or more branches of the curve intersect. The multiplicity in the case of a cusp comes from the coalescing of points which are ‘infinitely near’ to one another compared with the overlaying of points from two different branches of the curve in the nodal case. For an accessible discussion of curve singularities see F.Kirwan, Complex Algebraic Curves (Cambridge 1992). Weil employs the terminology of ‘simple’ and ‘multiple’ point which, although graphic, has lost favour in recent years. 18 A.Weil, Foundations of Algebraic Geometry, 2nd edn. (New York 1962), p. viii. The most recent treatments of intersection theory have returned, to some extent, to a more geometric approach. See, for example, W. Fulton, Intersection Theory (New York 1984). 19 The universal domain may therefore be considered as the field of rational functions in countably many indeterminates over the algebraic closure of the base field. 20 ibid., pp. 304 et seq. 21 Although this may sound a curious procedure it arises very naturally in certain number theoretic problems related to ‘Néron models’ of curves. 22 See B.Zilber and E.Hrushovski, ‘Zariski Geometries’, Bulletin of the American Mathematical Society (New Series), 28, (1993): 315–24, for a model theoretic version of this type of algebraic geometry which is independent of characteristic. 23 Not a cycle since the point is singular. 24 The subtleties of Weil’s approach have not always found their way into textbook expositions of this material. The discussion in the text applies to both affine and ‘abstract’ varieties. 25 Published together as Courbes algébriques et variétés abeliennes, (Markgröningen 1948). 26 ibid., p. 87. 27 A.Weil, Oeuvres scientifiques, vol. I, pp. 399–410. 28 An overview of Grothendieck’s career and publications can be found in the foreword to The Grothendieck Festschrift, vol. I (Berlin 1990).

NOTES 177

29 See S.MacLane, Categories for the Working Mathematician, (New York 1971), as a standard reference for category theory. The history of the development of algebraic topology is traced in great detail in J. Dieudonné, A History of Algebraic Topology 1900–1960 (Berlin 1990). 30 For a thorough discussion of the ‘functor of points of a scheme’ see D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Mathematics 1358, (New York 1988), pp. 155–68. To be more precise, the functor in question is representable by a ‘locally ringed space’. 31 am forced to be vague about this concept of ‘covering’ and ‘overlapping’ (or ‘glueing’ or ‘patching’ as it is sometimes called) as a fuller development would require a discussion of a number of algebraic concepts which would take us too far afield. For details see M.Demazure and P.Gabriel, Introduction to Algebraic Geometry and Algebraic Groups, (Amsterdam 1980), pp. 11–16. 32 The standard discussion of schemes is A.Grothendieck and J.Dieudonné, Elements de géométrie algébrique, vol. I (New York 1971) (known as EGA I), while topoi and sheaf categories are discussed in A. Grothendieck, M. Artin and J.-L.Verdier, Séminaire de Géométrie Algébrique, Lecture Notes in Mathematics 269, 270, 305 (New York 1972, 1973) (known as SGA IV 1–3). 33 It is of course possible to produce this type of construction using Weil’s language as well. My point here is that these relative notions are central to the structure of Grothendieck’s argument. 34 In particular, the co-ordinate rings are now allowed to have zero-divisors so that Weil’s restriction to regular extensions is no longer required. 35 EGA I, pp. 31–4. 36 This relativity also extends to the set theoretical foundations of the theory. Grothendieck introduces a set theoretic ‘universe’ which provides the sets used in his constructions and then studies those properties which remain invariant under a change of universe! See SGA IV, vol. 1, exp. 1. 37 While the geometric properties of space are undetermined the nature of the Euclidean enterprise is rigidly controlled. 38 There has been much discussion of the ‘purposes’ for which Euclid introduces various concepts or carries out certain portions of his investigation. From the point of view of ‘deductive structure’, as Mueller points out, there are many culs-de-sac in the logical structure of the Elements and it is thus tempting to suppose that certain subjects were included or treated in depth for external philosophical, pedagogical or similar reasons. The reading given here is by no means a complete response to this type of question, but may perhaps be useful in suggesting additional avenues for investigation. 39 Aristotle’s discussion of the role of ‘soul’ in psychology contains many parallels. R.McKeon, ‘De anima: Psychology and Science’, Journal of Philosophy, 21, (1930): 673–90. 40 See the discussion in Heath, vol. 3, following Propositions 1–3 of Book XI for what follows.

178 NOTES

6 Generality in contemporary mathematics 1 ‘As a final affirmation that we have plunged into the icy stream of modern mathematics, hardly a picture appears.’ M.Spivak, A Comprehensive Introduction to Differential Geometry, vol. 5 (Publish or Perish 1975), p. 385. 2 A.Weil, Basic Number Theory (New York 1967). 3 A.Weil, Oeuvres scientifiques, vol. I, pp. 187–240. 4 A useful treatment emphasizing the parallels discussed in the text can be found in H. Hasse, Number Theory (New York 1962). I have also benefitted from an unpublished lecture of B.Mazur entitled ‘From Algebra to Geometry’ in writing this section. 5 See, for example, A.Weil, Basic Number Theory and the well-known thesis of J.Tate published as Chapter 15 of I.Cassels and A.Frohlich, Algebraic Number Theory, (London 1967). A.Beilinson has shown how to extend the notion of adèles to an arbitrary (noetherian) scheme but only in so far as the finite primes are concerned. See A.Beilinson, ‘Higher Dimensional Adéles’ Functional Analysis and its Applications, 14, (1980): 34–5. This area is undergoing rapid transformation through the introduction of ideas from Conformal Field Theory. 6 See C.Soulé, D.Abramovitch, J.-F.Burnol and J.Kramer, Lectures on Arakelov Geometry (Cambridge 1992), and C.Soulè and H.Gillet, ‘Arithmetic Riemann-Roch’, Inventiones Mathematicae, 110, (1992): 473– 545. 7 A.Weil, Oeuvres scientifiques, vol, 1, pp. 250–1 (translation mine). As to why he was in prison at the time, Weil tells the story himself in A. Weil, Souvenirs d’apprentissage, (Berlin 1991), ch. VI. 8 He remarks that this is intimately connected to the philosophy of the Bhaghavad Gita. Weil has emphasized the role of Indian poetry and philosophy in his scientific development. This aspect of his thought awaits a competent commentator. 9 An introduction to this circle of ideas is given in Mac Lane, Categories for the Working Mathematician, pp. 40–5. 10 To complete the definition, one also requires that where two such neighbourhoods overlap, the map which relates the product structure given by the first neighbourhood to that given by the second be linear (and continuous, smooth etc.). 11 The classical treatment is N.Steenrod, Fiber Bundles (Princeton 1951). 12 This is an example of a so-called ‘comma category’. Mac Lane, Categories, pp 46–8. 13 See A.Vistoli, ‘Intersection Theory on Stacks’, Inventiones Mathematicae, 97, (1989): 613–70. The appendix to this article contains a useful introduction to the general notion of stack and, more specifically, to ‘algebraic stacks’ which are a natural generalization of schemes.

NOTES 179

14 A.Grothendieck, ‘Sur quelques points d’algèbre homologique’, Tohôku Mathematics Journal, 9, (1957): 119–221. 15 See P.May, Simplicial Objects in Algebraic Topology, (Chicago 1967), for details. 16 Although the idea is often credited to Grothendieck, the first detailed treatment is G.Segal, ‘Classifying Spaces of Categories’, Publications de l’ Institut des Hautes Etudes Scientifiques, 34, (1968): 105–12. 17 D.Quillen, Homotopical Algebra, Lecture Notes in Mathematics 41, (New York 1967). 18 That is, the ‘homotopy categories’ (a version of derived category applicable in this context) are equivalent. If one considers only the ‘fibrant’ simplicial objects then the equivalence is more direct. See May, Simplicial Objects. 19 Note the resemblance to simplicial objects. 20 P.Deligne, ‘Le Determinant de la cohomologie’, in K.Ribet (ed.) Current Trends in Arithmetical Algebraic Geometry, Contemporary Mathematics 67 (New York 1987). See also SGA IV, vol. 3, exp. XVIII. 21 Technically this construction applies only to exact categories. I do not know whether it is possible to extend it following Waldhausen’s extension of algebraic K-theory to a broader collection of categories. 22 A.Grothendieck, Esquisse d’un programme (unpublished notes 1984). 23 ibid. pp.43–4. 24 The principal rubrics for this work include Tannakian categories, YangBaxter equations and braided tensor categories. 25 I have in mind here something like Dewey’s structuring of action with an ‘end-in-view’.

Conclusion 1 In more dyspeptic moments, however, I would refer those who would criticize the lack of such reference to Tovey’s report of Brahms’ remarks concerning critics who made much of the resemblances between his first symphony and Beethoven’s ninth. 2 This is not fixed once and for all. The examples of texts, thought of as ‘dead’ by one generation, which have supplied mathematical insights to later generations are legion. 3 ‘
göttlichen Geschlechte
’. R.Dedekind, Werke, vol. 3, p. 489 (letter to H.Weber of 24 January 1888). 4 ‘
schöpferische Kraft
’. Ibid.

180

Index

Abel, N. 102 abelian extension 101 abelian variety 120 absolute irreducibility 127 adèles 146 algebraic integer 78 algebraic number 78 algorithm 56 Arakelov theory 147-149 arithmetic algebraic geometry 96, 103, 119-121 axiom of Archimedes 67, 70 Baker, A. 121 Bessel, K. 75 Betti numbers 129 bi-quadratic reciprocity 76 Bourbaki, N. 79 Bunte Bemerkungen 99 Cantor, G. 65, 75 cardinal numbers 97 category 129-132 Cauchy sequence 145 circumscribed 52, 140, 141 chain 126 champs 152 class field theory 81, 99, 101 class of an equation 27 cogito 19 complex number 76 composite numbers 66 composition of forms 77 congruence of triangles 42 consistency proof 34

constructivism 95 coordinates 52 correlative correspondence 122 covariance 12 cyclotomic numbers 76-89, 99 cycle 127 Data 12 degree of a field extension 109 Deligne, P. 129, 154 derived category 152 Desargues 34 differential geometry 43, 144, 147 diophantine geometry 120 Dirichlet, P.G.L. 76-81, 111, 122, 144 Dirichlet—Dedekind 81-96, 109-113 discriminant 83, 86, 92, 93, 111 divisions of a problem 24 divisor 96, 98 Edwards, H.M. 78, 87, 96, 98 Eisenstein, G. 103 elliptic curves with complex multiplication 96, 102 equimultiples 58-61, 63, 67 even numbers 66 Euclidean algorithm 67-67 Faltings, G. 121 fibered categories 151, 152 flat 8 Foundations of Algebraic Geometry 124-128 fourth proportional 61 181

182 INDEX

function theory 43 galois group 92, 94, 102, 112 Gauss, C.F. 74-85, 98, 111 general position 124 generator of an ideal 91 generic point 128 genus of a curve 120 geometrical sophisms 41 Giraud, J. 155 Göttingen 75, 76 group theory 43 Grundzüge 96-98 Heath, Sir T. 1, 4, 59, 71, 137, 137 heights 120 Hensel, K. 104 higher congruences 81 hyper-complex numbers 84 hypothesis 69, 134 ideal numbers 77-91, 97 ideal of a ring 110 Illusie, L. 154 index of an algebraic number 83, 111 inessential discriminant divisor 99 inscribed 52, 140, 141 intersection theory 122 intermediate value theorem 97 intuitionism 95 irrational number 61, 69, 79, 87 irreducible 126, 127 j-function 113 Jordan, C. 102 Jugendtraum theorem 96, 101, 105, 112 Kant, I. 74 Kronecker-Weber theorem 96, 101, 112 Kummer, E.E. 76-89, 94-99 Langlands, R. 96, 113 Lefschetz principle 125 Leibniz. G.F. 74 L-function 103, 122

limit theorem 96 linear transformation 109 measure theory 4 Menaechmus 20, 21 metric 4 Milnor, J. 153 Minkowski, H. 79, 101 module 85, 92, 97 Mordell, L.J. 119, 120 Mordell-Weil theorem 119-121 Mueller, I. 61 nerve of a category 153 Newton, I. 74 Noether, E. 85, 92 odd numbers 66 o-minimal 104 p-adic numbers 81, 104, 145, 146 Pappus 25, 28 passive perfect imperative 15 parallelogrammatic areas 45 Pell’s equation 102 Picard category 154, 155 place 144 plane number 66 positive characteristic 122, 125 power basis 83 prime ideal 110 prime number 66 primitive root of unity 82 principal ideal 110 Proclus 20 projective geometry 120 Pythagoras 69 quadratic forms 77 quadratic reciprocity 69 Quillen, D. 152, 154 ramification 84-98 rank 121 rational integer 76 rational numbers 56, 79 rational transformation 120

INDEX 183

rationalitãts bereiche 98 real numbers 69 reciprocity law 76 Riemann, B. 75 Riemann-Roch theorem 144-147 Riemann sphere 145 ring 84 ring of integers of a number field 109-113 scheme 130-132 Segal, G. 153 Severi, E. 122 sheaf 130 Siegel, C.L. 120, 121 simplicial object 152, 153 Simson, R. 72 singular point 124 solid numbers 66 specialization 127 Speussipus 20, 21 Stetigkeit und irrationalen Zahlen 80 Sturm’s theorem 97, 104 superposition 42, 43 théorème de decomposition 120 Theorie der algebraischer Funktionen einer verandlicher 79 theta function 41 topology 76 topos 131 2-category 150-152 unique factorization 76-89 universal domain 125 valuation 145 vector bundle 150 Verdier, J-L 152 Was sind und was sollen die Zahlen 86 Weber, H. 79, 112 Weierstrass, K. 75 Weil cohomology theory 122 Weil, S. 148 Weyl, H. 104

Zahlbericht 78 Zariski, O. 124 zeta function 121