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AN INTRODUCTION TO
MATHEMATICS
CHAPTER I
THE ABSTRACT NATURE OF MATHEMATICS
The study of mathematics is apt to commence in disap-
pointment. The important applications of the science, the
theoretical interest of its ideas, and the logical rigour of its
methods, all generate the expectation of a speedy introduc-
tion to processes of interest. We are told that by its aid the
stars are weighed and the billions of molecules in a drop of
water are counted. Yet, like the ghost of Hamlet’s father,
this great science eludes the efforts of our mental weapons
to grasp it—“ ’Tis here, ’tis there, ’tis gone”—and what we
do see does not suggest the same excuse for illusiveness as
sufficed for the ghost, that it is too noble for our gross meth-
ods. “A show of violence,” if ever excusable, may surely
be “offered” to the trivial results which occupy the pages of
some elementary mathematical treatises.
The reason for this failure of the science to live up to its
reputation is that its fundamental ideas are not explained to
the student disentangled from the technical procedure which
has been invented to facilitate their exact presentation in
particular instances. Accordingly, the unfortunate learner
finds himself struggling to acquire a knowledge of a mass
of details which are not illuminated by any general concep-
NATURE OF MATHEMATICS
tion. Without a doubt, technical facility is a first requisite
for valuable mental activity: we shall fail to appreciate the
rhythm of Milton, or the passion of Shelley, so long as we
find it necessary to spell the words and are not quite certain
of the forms of the individual letters. In this sense there is no
royal road to learning. But it is equally an error to confine
attention to technical processes, excluding consideration of
general ideas. Here lies the road to pedantry.
The object of the following Chapters is not to teach math-
ematics, but to enable students from the very beginning of
their course to know what the science is about, and why it
is necessarily the foundation of exact thought as applied to
natural phenomena. All allusion in what follows to detailed
deductions in any part of the science will be inserted merely
for the purpose of example, and care will be taken to make
the general argument comprehensible, even if here and there
some technical process or symbol which the reader does not
understand is cited for the purpose of illustration.
The first acquaintance which most people have with
mathematics is through arithmetic. That two and two make
four is usually taken as the type of a simple mathematical
proposition which everyone will have heard of. Arithmetic,
therefore, will be a good subject to consider in order to
discover, if possible, the most obvious characteristic of the
science. Now, the first noticeable fact about arithmetic is
that it applies to everything, to tastes and to sounds, to ap-
ples and to angels, to the ideas of the mind and to the bones
of the body. The nature of the things is perfectly indifferent,
INTRODUCTION TO MATHEMATICS
of all things it is true that two and two make four. Thus
we write down as the leading characteristic of mathematics
that it deals with properties and ideas which are applica-
ble to things just because they are things, and apart from
any particular feelings, or emotions, or sensations, in any
way connected with them. This is what is meant by calling
mathematics an abstract science.
The result which we have reached deserves attention. It
is natural to think that an abstract science cannot be of
much importance in the affairs of human life, because it has
omitted from its consideration everything of real interest. It
will be remembered that Swift, in his description of Gul-
liver’s voyage to Laputa, is of two minds on this point. He
describes the mathematicians of that country as silly and
useless dreamers, whose attention has to be awakened by
flappers. Also, the mathematical tailor measures his height
by a quadrant, and deduces his other dimensions by a rule
and compasses, producing a suit of very ill-fitting clothes.
On the other hand, the mathematicians of Laputa, by their
marvellous invention of the magnetic island floating in the
air, ruled the country and maintained their ascendency over
their subjects. Swift, indeed, lived at a time peculiarly un-
suited for gibes at contemporary mathematicians. Newton’s
Principia had just been written, one of the great forces which
have transformed the modern world. Swift might just as well
have laughed at an earthquake.
But a mere list of the achievements of mathematics is an
unsatisfactory way of arriving at an idea of its importance.
NATURE OF MATHEMATICS
It is worth while to spend a little thought in getting at the
root reason why mathematics, because of its very abstract-
ness, must always remain one of the most important topics
for thought. Let us try to make clear to ourselves why ex-
planations of the order of events necessarily tend to become
mathematical.
Consider how all events are interconnected. When we see
the lightning, we listen for the thunder; when we hear the
wind, we look for the waves on the sea; in the chill autumn,
the leaves fall. Everywhere order reigns, so that when some
circumstances have been noted we can foresee that others will
also be present. The progress of science consists in observing
these interconnections and in showing with a patient ingenu-
ity that the events of this evershifting world are but examples
of a few general connections or relations called laws. To see
what is general in what is particular and what is permanent
in what is transitory is the aim of scientific thought. In the
eye of science, the fall of an apple, the motion of a planet
round a sun, and the clinging of the atmosphere to the earth
are all seen as examples of the law of gravity. This possibility
of disentangling the most complex evanescent circumstances
into various examples of permanent laws is the controlling
idea of modern thought.
Now let us think of the sort of laws which we want in order
completely to realize this scientific ideal. Our knowledge of
the particular facts of the world around us is gained from
our sensations. We see, and hear, and taste, and smell, and
feel hot and cold, and push, and rub, and ache, and tingle.
INTRODUCTION TO MATHEMATICS
These are just our own personal sensations: my toothache
cannot be your toothache, and my sight cannot be your sight.
But we ascribe the origin of these sensations to relations
between the things which form the external world. Thus
the dentist extracts not the toothache but the tooth. And
not only so, we also endeavour to imagine the world as one
connected set of things which underlies all the perceptions of
all people. There is not one world of things for my sensations
and another for yours, but one world in which we both exist.
It is the same tooth both for dentist and patient. Also we
hear and we touch the same world as we see.
It is easy, therefore, to understand that we want to de-
scribe the connections between these external things in some
way which does not depend on any particular sensations,
nor even on all the sensations of any particular person. The
laws satisfied by the course of events in the world of external
things are to be described, if possible, in a neutral universal
fashion, the same for blind men as for deaf men, and the
same for beings with faculties beyond our ken as for normal
human beings.
But when we have put aside our immediate sensations,
the most serviceable part—from its clearness, definiteness,
and universality—of what is left is composed of our general
ideas of the abstract formal properties of things; in fact,
the abstract mathematical ideas mentioned above. Thus it
comes about that, step by step, and not realizing the full
meaning of the process, mankind has been led to search for
a mathematical description of the properties of the universe,
NATURE OF MATHEMATICS
because in this way only can a general idea of the course of
events be formed, freed from reference to particular persons
or to particular types of sensation. For example, it might be
asked at dinner: “What was it which underlay my sensation
of sight, yours of touch, and his of taste and smell?” the an-
swer being “an apple.” But in its final analysis, science seeks
to describe an apple in terms of the positions and motions of
molecules, a description which ignores me and you and him,
and also ignores sight and touch and taste and smell. Thus
mathematical ideas, because they are abstract, supply just
what is wanted for a scientific description of the course of
events.
This point has usually been misunderstood, from being
thought of in too narrow a way. Pythagoras had a glimpse
of it when he proclaimed that number was the source of all
things. In modern times the belief that the ultimate expla-
nation of all things was to be found in Newtonian mechanics
was an adumbration of the truth that all science as it grows
towards perfection becomes mathemematical in its ideas.