JOHN L. BELL Department of Philosophy, University of Western Ontario THE ART OF THE INTELLIGIBLE An Elementary Survey of Mathematics in its Conceptual Development To my dear wife Mimi The purpose of geometry is to draw us away from the sensible and the perishable to the intelligible and eternal. Plutarch TABLE OF CONTENTS FOREWORD page xi ACKNOWLEDGEMENTS xiii CHAPTER 1 NUMERALS AND NOTATION 1 CHAPTER 2 THE MATHEMATICS OF ANCIENT GREECE 9 CHAPTER 3 THE DEVELOPMENT OF THE NUMBER CONCEPT 28 THE THEORY OF NUMBERS Perfect Numbers. Prime Numbers. Sums of Powers. Fermat’s Last Theorem. The Number π . WHAT ARE NUMBERS? CHAPTER 4 THE EVOLUTION OF ALGEBRA, I 53 Greek Algebra Chinese Algebra Hindu Algebra Arabic Algebra Algebra in Europe The Solution of the General Equation of Degrees 3 and 4 The Algebraic Insolubility of the General Equation of Degree Greater than 4 Early Abstract Algebra 70 CHAPTER 5 THE EVOLUTION OF ALGEBRA, II 72 Hamilton and Quaternions. Grassmann’s “Calculus of Extension”. Finite Dimensional Linear Algebras. Matrices. Lie Algebras. CHAPTER 6 THE EVOLUTION OF ALGEBRA, III 89 Algebraic Numbers and Ideals. ABSTRACT ALGEBRA Groups. Rings and Fields. Ordered Sets. Lattices and Boolean Algebras. Category Theory. CHAPTER 7 THE DEVELOPMENT OF GEOMETRY, I 111 COORDINATE/ALGEBRAIC/ANALYTIC GEOMETRY Algebraic Curves. Cubic Curves. Geometric Construction Problems. Higher Dimensional Spaces. NONEUCLIDEAN GEOMETRY CHAPTER 8 THE DEVELOPMENT OF GEOMETRY, II 129 PROJECTIVE GEOMETRY DIFFERENTIAL GEOMETRY The Theory of Surfaces. Riemann’s Conception of Geometry. TOPOLOGY Combinatorial Topology. Point-set topology. CHAPTER 9 THE CALCULUS AND MATHEMATICAL ANALYSIS 151 THE ORIGINS AND BASIC NOTIONS OF THE CALCULUS MATHEMATICAL ANALYSIS Infinite Series. Differential Equations. Complex Analysis. CHAPTER 10 THE CONTINUOUS AND THE DISCRETE 173 CHAPTER 11 THE MATHEMATICS OF THE INFINITE 181 CHAPTER 12 THE PHILOSOPHY OF MATHEMATICS 192 Classical Views on the Nature of Mathematics. Logicism. Formalism. Intuitionism. APPENDIX 1 THE INSOLUBILITY OF SOME GEOMETRIC CONSTRUCTION PROBLEMS 209 APPENDIX 2 THE GÖDEL INCOMPLETENESS THEOREMS 214 APPENDIX 3 THE CALCULUS IN SMOOTH INFINITESIMAL ANALYSIS 222 APPENDIX 4 THE PHILOSOPHICAL THOUGHT OF A GREAT MATHEMATICIAN: HERMANN WEYL 229 BIBLIOGRAPHY 234 INDEX OF NAMES 236 INDEX OF TERMS 238 FOREWORD My purpose in writing this book is to present an overview—at a fairly elementary level—of the conceptual evolution of mathematics. As will be seen from the Table of Contents, I have adhered in the main to the traditional tripartite division of the subject into Algebra (including the Theory of Numbers), Geometry, and Analysis. I have attempted to describe, in roughly chronological order, what are to me some of the most beautiful, and—if I am not entirely misguided—some of the most significant developments in each of these domains. In this spirit I have also included brief accounts of mathematical notation, ancient Greek mathematics, set theory, and the philosophy of mathematics. The Appendices contain short expositions of topics which are particularly dear to my heart and which I hope my readers (if any) will take to theirs. My approach here—a curious mix, admittedly, of the chronological and the expository—is the result, not entirely of my whim, but also of having spent a number of years of lecturing on these topics to undergraduates. I should point out that the use of the word “intelligible” in the book’s title is intended to convey a double meaning. First, of course, the usual one of “comprehensible” or “capable of being understood.” But the word also has an older meaning, namely, “capable of being apprehended only by the intellect, not by the senses”; in this guise it serves as an antonym to “sensible”. It is precisely with this signification that Plutarch uses the word in the epigraph I have chosen. While the potential intelligibility of mathematics in this older sense is hardly to be doubted, I can only hope that my book conveys something of that intelligibility in its more recent connotation. ACKNOWLEDGEMENTS It was Ken Binmore—my colleague for many years at the London School of Economics—who proposed in the 1970s that I develop a course of lectures under the catch-all title “Great Ideas of Mathematics”. Twenty years or so later Rob Clifton—my erstwhile (now sadly deceased) colleague at the University of Western Ontario— suggested that I give the same course here, a stimulus which led to the expansion and polishing of the somewhat primitive notes I had prepared for the original course, and which also had the effect of emboldening me to turn them into the present book. In the last analysis it must be left to readers of the book to judge whether my two confrères are to be applauded for their encouragement of my efforts, but for my part I am happy to acknowledge my debt to them. I would also like to thank Elaine Landry for suggesting that I give the course of lectures on the philosophy of mathematics which ultimately led to the writing of Chapter 12 of the book. I am grateful to Alberto Peruzzi for his helpful comments on an early draft of the manuscript, and to Max Dickmann for his scrutiny of the hardback edition of the book, leading to the discovery of a number of errata (which are listed on a separate sheet in the present edition). To my wife Mimi I tender special thanks both for inscribing the Chinese numerals in Chapter 1 and for her heroic efforts in reading aloud to me the entire typescript (apart, of course, from the formulas). I must also acknowledge the authors of the many books—listed in the Bibliography—which I have used as my sources. Finally, I would like to record my gratitude to Rudolf Rijgersberg at Kluwer for the enthusiasm and efficiency he has brought to the project of publishing this book. CHAPTER 1 NUMERALS AND NOTATION FOR CENTURIES MATHEMATICS WAS DEFINED as “the science of number and magnitude”, and while this definition cannot nowadays be taken as adequate, it does, nevertheless, reflect the origins of mathematics with reasonable fidelity. Notions related to the concepts of number and magnitude can be traced back to the dawn of the human race. Indeed some animal species—whose origins antedate those of humanity by millions of years—behave in such a way as to reveal a rudimentary mathematical sense: experiments with crows, for example, have shown that birds can distinguish among sets containing up to four elements. At any rate, mathematical thinking has long played a role in the everyday practical life of human beings. The origin of the number concept, in particular, would seem to lie in our remote ancestors’ grasping the idea of plurality, and seeing that pluralities or collections of things can be both matched and compared in size. For example, the hands can be matched with the feet or the eyes, the fingers with the toes, but not the feet with the fingers. The realization that matchability of hands, feet, eyes, and any other pair of objects is independent of their nature must have provided a crucial stimulus for the emergence of the idea of number. That small numbers such as two and four played an early and important role in human thought is shown by their special position in the grammar of certain languages, for example Greek, in which a distinction is made between one, two, and more than two, and Russian, which uses one noun case with numbers up to four, and a different one for larger numbers. Numbers are assigned to collections by means of the process of counting, that is, the procedure of matching the elements of a collection successively with the ascending sequence of numbers, or number names. The recognition that the procedure of counting “one, two, three, four, ...” can be performed intransitively, in other words, that when counting it is not necessary to be actually counting something, is likely to have been instrumental in establishing the universality of the number concept. Indeed, it has been suggested that the art of counting arose in connection with primitive religious ritual and that the counting or ordinal aspect of number preceded the emergence of the quantitative or cardinal aspect. Whatever its origin, the procedure of counting naturally imposes an order on numbers, and it must have been grasped very early on that this order corresponds faithfully to the relative sizes of the collections that numbers are used to count. As the awareness of number developed, it became necessary to express or represent the idea by means of signs. The earliest and most immediate mode of representing numbers is by means of the fingers, which conveniently enable numbers up to ten to be 2 CHAPTER 1 represented. For larger numbers heaps of pebbles were used, piled in groups of five, each corresponding to the number of fingers on the hand. Indeed our very term “calculate” derives from the Latin word calculus meaning “small stone”. Counting by fives and tens—and, in some cultures, twenties—became standard practice, largely displacing the earlier systems of counting by twos and threes. Thus arose the idea of a base or scale for counting, in which some number b is selected, names for (some of the) numbers 1,..., b are assigned, and names for numbers larger than b are formulated as combinations of these. It was observed by Aristotle that the customary choice of base 10 is merely the result of the accidental fact that human beings happen to possess five fingers on each of two hands: from a strictly mathematical point of view it is somewhat regrettable that our ancestors did not possess a composite number of fingers, such as four or six, on each hand, rather than the awkward prime number five.