Table Of ContentENCOUNTER WITH
MATHEMATICS
LARS GARDING
ENCOUNTER WITH
MATHEMATICS
SPRINGER-VERLAG
NEW YORK HEIDELBERG BERLIN
Lars Garding
Mat. Inst. Fack 725
Lund 7, Sweden
Library of Congress Cataloging in Publication Data
Garding, Lars, 1919-
Encounter with mathematics.
Includes index.
I. Matlrematics-1961- I. Title. II. Series.
QA37.2.G28 510 76-54765
All rights reserved.
No part of this book may be translated or reproduced in any
form without written permission from Springer-Verlag.
© 1977 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1s t edition 1977
9 8 7 6 5 432 I
ISBN -13: 978-1-4615-9643-1 e-ISBN-13: 978-1-4615-9641-7
DOl: 10.1007/978-1-4615-9641-7
PREFACE
Trying to make mathematics understandable to the general public is a very
difficult task. The writer has to take into account that his reader has very
little patience with unfamiliar concepts and intricate logic and this means
that large parts of mathematics are out of bounds.
When planning this book, I set myself an easier goal. I wrote it for those
who already know some mathematics, in particular those who study the
subject the first year after high school. Its purpose is to provide a
historical, scientific, and cultural frame for the parts of mathematics that
meet the beginning student. Nine chapters ranging from number theory to
applications are devoted to this program. Each one starts with a historical
introduction, continues with a tight but complete account of some basic
facts and proceeds to look at the present state of affairs including, if
possible, some recent piece of research. Most of them end with one or two
passages from historical mathematical papers, translated into English and
edited so as to be understandable.
Sometimes the reader is referred back to earlier parts of the text, but the
various chapters are to a large extent independent of each other. A reader
who gets stuck in the middle of a chapter can still read large parts of the
others. It should be said, however, that the book is not meant to be read
straight through. It contains a wealth of material, some of it not considered
elementary, e.g. Hilbert's Nullstellensatz in the algebra chapter and the
Fourier inversion formula in the chapter on integration. These important
items were included because in both cases it was possible to give simple
and lucid proofs that fitted the context.
Three chapters are of a more general nature. The introductory one deals
with models and reality and the final one with the sociology, psychology,
and teaching of mathematics. There is also a chapter on the mathematics
of the seventeenth century providing a fuller historical background to
infinitesimal calculus. In an appendix there are a few words on terminol
ogy and notation and some advice on how to read and choose mathemati
cal texts.
A preliminary draft of the book was read by Karl Gustav Andersson
and Tomas Claesson, and Gunnar Blom read the probability chapter. The
final draft has been read by William F. Donoghue Jr, Tore Herlestam, and
Charles Halberg. I thank these six friends and critics for much valuable
advice.
April, 1977 Lars Garding
Lund
v
CONTENTS
1
MODELS AND REALITY 1
1.1 Models 1
1.2 Models and reality 5
1.3 Mathematical models 6
2
NUMBER THEORY 9
2.1 The primes 10
2.2 The theorems of Fermat and Wilson 13
2.3 The Gaussian integers 15
2.4 Some problems and results 19
2.5 Documents 21
3
ALGEBRA 23
3.1 The theory of equations 24
3.2 Rings, fields, modules, and ideals 30
3.3 Groups 40
3.4 Documents 53
4
GEOMETRY AND LINEAR ALGEBRA 57
4.1 Euclidean geometry 58
4.2 Analytical geometry 64
4.3 Systems of linear equations and matrices 71
4.4 Linear spaces 80
4.5 Linear spaces with a norm 86
4.6 Boundedness, continuity, and compactness 89
4.7 Hilbert spaces 93
4.8 Adjoints and the spectral theorem 98
4.9 Documents 102
5
LIMITS, CONTINUITY, AND TOPOLOGY 105
5.1 Irrational numbers, Dedekind's cuts, and
Cantor's fundamental sequences 106
VB
Contents
5.2 Limits of functions, continuity, open and closed sets III
5.3 Topology 116
5.4 Documents 122
6
THE HEROIC CENTURY 124
7
DIFFERENTIATION 134
7.1 Derivatives and planetary motion 135
7.2 Strict analysis 140
7.3 Differential equations 144
7.4 Differential calculus of functions of several variables 146
7.5 Partial differential equations 151
7.6 Differential forms 155
7.7 Differential calculus on a manifold 159
7.8 Document 166
8
INTEGRATION 168
8.1 Areas, volumes, and the Riemann integral 168
8.2 Some theorems of analysis 173
8.3 Integration in Rn and measures 188
8.4 Integration on manifolds 194
8.5 Documents 203
9
SERIES 205
9.1 Convergence and divergence 206
9.2 Power series and analytic functions 209
9.3 Approximation 215
9.4 Documents 219
10
PROBABILITY 222
10.1 Probability spaces 222
10.2 Stochastic variables 225
10.3 Expectation and variance 229
10.4 Sums of stochastic variables, the law of large numbers,
and the central limit theorem 231
10.5 Probability and statistics, sampling 233
10.6 Probability in physics 236
10.7 Document 238
viii
Contents
11
APPLICATIONS 240
11.1 Numerical computation 240
11.2 Construction of models 245
12
THE SOCIOLOGY, PSYCHOLOGY, AND
TEACHING OF MATHEMATICS 253
12.l Three biographies 253
12.2 The psychology of mathematics 257
12.3 The teaching of mathematics 259
APPENDIX 263
INDEX 265
ix
1
MODELS AND REALITY
l.l Models. The natural numbers. Celestial mechanics. Quantum
mechanics. Economics. Language. 1.2 Models and reality. 1.3
Mathematical models.
Trying to understand the world around him, man organizes his observa
tions and ideas into conceptual frames. These we shall call models. The
insight gotten by applying logic to the concepts of a model will be called
its theory. Mathematical models are logically coherent and have extensive
theories. Others may be less strict but no less useful.
In the exact sciences, the validity of models is tested by logic and by
experiment. This makes it necessary to distinguish very clearly between the
model and the part of the outside world that it is supposed to represent.
This principle is now current in many branches of science. When applied
in a general way it puts human thought into an interesting perspective.
Part of this chapter deals with man's relations to the models of the world
that he himself has created. It starts with short descriptions and evalua
tions of some important models and ends with a survey of some mathe
matical models and their mutual relations.
1.1 Models
The natural numbers
The simplest mathematical model is the set of natural numbers
I, 2, 3, . .. . It is used for counting objects when all the properties of
objects are disregarded except their number. The natural numbers appear
in all languages. Some have names for more numbers than others, but
there are always numbers that are so large that they have no names. This
phenomenon is perhaps man's first encounter with infinity. In ancient
times it led to serious questions like these: are there numbers so large that
they cannot be counted? Or, in a more concrete setting: are the grains of
sand on the earth uncountable? The second question was answered by
Archimedes in a book called The Sand Reckoner (200 B.C.). He displayed a
series of rapidly growing numbers and could show by some estimates of
volume that some of these numbers were larger than the number of grains
of sand on the earth and even in the solar system. We see here how the
model of natural numbers answers a concrete question about the outside
Models and reality
world. Abstraction has proved its worth. The situation is illustrated by the
left part of Figure 1.1. The ragged contour on the left means that we have
cut out a piece of the real world that can have properties without
counterparts in the model. The straight lines and right angles of the model
are supposed to illustrate its schematic character.
•
questions questions
model
answers answers
Figure I. I The triplet Reality-model-theory.
Let us now add the operation of multiplication to our model, the
natural numbers. It then acquires a lot of very interesting properties. Some
experimentation with multiplication shows that certain numbers are prod
ucts of other numbers greater than 1, e.g., 20 = 2 X 2 X 5, while others, e.g.,
5, do not have this property. Those of the second kind are called prime
numbers or primes. The first primes are 2,3, 5, 7, 11, 13, 17. It is easy to
continue the series but the amount of necessary checking increases very
quickly when we come to large numbers. Under these circumstances, the
following question presents itself: is there a finite or infinite number of
primes? A simple and ingenious reasoning to be found in Euclid's Elements
(270 B.C.) and explained in the next chapter provides the answer: the
number of primes is infinite. We have here an example of a question put in
the model that can be answered by theory, i.e., logical reasoning about the
model. This is illustrated by the right part of Figure 1.1. The ragged
contour of the theory indicates that it is not determined completely by the
model. In our case it contains a host of theorems about the natural
numbers, e.g., solvability and nonsolvability of certain equations, the
distribution of primes, etc. The size and the power of the theory depends
among other things on the ability of the mathematicians who created it.
Our three-part figure shall now serve to illustrate a number of important
models other than the natural numbers.
Celestial mechanics
The part of reality that is to be analyzed consists of astronomical
observations of the positions of the earth, the planets, and the sun at
different times. In the model, these celestial bodies correspond to point
sized objects with masses that attract each other according to Newton's
2
1.1 Models
law of gravitation. Each object attracts every other one with a force whose
size is proportional to the product of the two masses and inversely
proportional to the square of the distance. This force is directed towards
the attracting object. The movement is such that the product of the mass
and the acceleration of an object equals the attractive force. The theory
consists of mathematical propositions about the nature of such move
ments. It turns out in particular that the positions and the velocities of the
objects at one time determine the subsequent motion uniquely. This model
and its theory were created by Newton in the seventeenth century. It
answers a multitude of astronomical questions. The orbits and masses of
the planets can be computed with great accuracy. Small deviations from
the predictions have led to the discovery of new planets. The orbits of
artificial satellites are predicted with the aid of this model. The theory has
grown continuously from the time of Newton to the present. Celestial
mechanics has been an unparalleled success and made a deep impression
on the philosophy of the eighteenth and nineteenth centuries.
Quantum mechanics
The problem is to analyze radiation from atoms, recorded as tracks and
spectral lines on photographic plates. The model is a variant of celestial
mechanics with objects corresponding to the atomic nuclei and the elec
trons, but the relatively close intuitive connection between the real objects
and those of the model is lost. Certain objects of the model must be
interpreted both as waves and as particles. The insight that led to the
quantum-mechanical model comes, to a large extent, from the concepts
that were fruitful in the theory of celestial mechanics, e.g., mass, energy,
and momentum. The photographic plates do not by themselves provide
much guidance. The theory of this model consists, among other things, of
propositions about Hilbert space, an infinite-dimensional analogue of
Euclidean space. Classical quantum mechanics has been very successful in
the sense that the radiation frequencies and their dependence on electro
magnetic fields are predictable from very few data. One property of the
model, the complementarity principle, has gained a certain status in
philosophy.
Small discrepancies between computed and observed frequencies have
led to a more sophisticated model which takes into account Einstein's
relativity theory, and in which the electromagnetic field is quantized in the
same way as the model of the atom is a quantization of the model of
planetary motion. This new model has been less successful. There were and
still are unsolved difficulties with its theory. The embarrassing fact is that
there is no model of this kind with a consistent theory that also has
interesting applications. On the other hand, it is possible in the present
models to predict certain facts from others. It is hard to guess what is
going to happen next. Is the model going to change or is it the theory that
will expand?
3
