Table Of ContentJOHN 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.
Heaps of pebbles, lacking ready portability as well as the requisite stability for
prolonged storage of numerical information, came to be replaced in prehistory by less
ephemeral and more portable devices, such as notches in a piece of wood or bone. The
oldest known example of such a device is a tally stick dating from paleolithic times
found in Moravia in 1937. It is the radius bone of a young wolf on which are incised 55
notches, arranged in two series, with 25 in the first and 30 in the second, both
assembled in groups of five. The existence of these and a few other, similar artifacts,
being more than thirty thousand years old, show that the use of the number concept,
and, in particular, that of the number five, long antedates such technological advances
as metal smelting or wheeled vehicles, and even the development of written language.
This latter fact suggests that the appearance of number signs preceded that of number
words, a plausible claim in any case since primitive signs for numbers such as notches
or strokes possess a visual immediacy which has no counterpart in speech. The primacy
of the sign or symbol in mathematics has persisted until the present day.
The earliest numerals or formal signs for numbers of which we possess definite
record appeared in Egypt around 3400 B.C. The Egyptian system of hieroglyphic
numerals employed strokes for numbers below ten, and special symbols for powers of
ten, for example:
∩ ∩ ∩ ∩
4 10 (a heel bone) 12 20
100 ? (a scroll) 200 ? ? 1000 (a lotus flower)
There was even a symbol for 10000000 in the form of an astonished man or a god
supporting the universe.
Numerals appeared in Egypt long before the first known numerical inscriptions in
India (3rd century B.C.), China (3rd century B.C.) and Crete (1200 B.C.). Egyptian
numerals constitute what is known as a simple grouping system. In such systems some
number b—called a base or scale (10 in the Egyptian, as in most other cases)—is
chosen and the powers b, b2 , b3, ... of b treated as units—the higher units. Symbols
for the latter, as well as for 1, are introduced and arbitrary numbers then represented by
