WHAT IS SCIENCE?

WHAT IS SCIENCE?

BY

NORMAN CAMPBELL

SG.D., F.INST.P.

METHUEN & CO. LTD.

36 ESSEX STREET W.C,

LONDON

FtW Published in 1921

TO

THE MEMORY OF

FREDERICK WILLIAM MOORMAN

SOMETIME PRESIDENT OF
THE LEEDS AND DISTRICT BRANCH OF

THE WORKERS' EDUCATIONAL ASSOCIATION

TO WHOSE INSPIRATION
THIS BOOK OWES ITS ORIGIN

458024

PREFACE

little book is written with the hope of
encouraging the study of science in the classes
of the Workers' Educational Association. In
spite of some splendid successes — notably the biology
classes of Mr. Norman Walker in the Yorkshire district —
science does not receive its full share of attention. Science
is the characteristic product of modern thought in the
realm of pure learning ; and yet there is a danger that
the W. E. A., which stands for new ideas, will become
the last stronghold of the reactionary doctrine that
science and culture are antagonistic.

Accordingly, my object has been to explain what are
the aims and objects of science and what kind of satis-
faction can be derived from its study. I have tried to
draw attention to those aspects of its more abstruse
departments that may be expected to appeal to men
and women of wide intellectual sympathies. The book
does not pretend to be " popular " or to provide an easy
hour's reading ; for all experience shows that mere
difficulties of thought are no bar to success in adult
education ; the enthusiasm of a leader is all that is
necessary to sustain interest. No writer can hope to
get into as close touch with his readers as a speaker with
his audience, and unless leaders can be found to treat
science in the spirit suggested, my efforts must necessarily
fail. But perhaps my efforts will help some who would
not otherwise have undertaken the task.

Since I have no object but to lead readers to the
systematic study of some special branch of science, and

vii

viii WHAT IS SCIENCE?

do not desire that they should confine their attention
to the generalities with which this book is concerned,
no references are given to more detailed works covering
the same ground. But perhaps it should be remarked
that the subjects discussed, though concerned with
science, are not part of science ; and that, accordingly,
there is much more difference of opinion about some of
them than there would be about subjects more strictly
scientific. Since it is my object to arouse interest
rather than to convey information, I have not hesitated
sometimes to assert dogmatically what others, equally
qualified to judge, would vehemently deny.

CONTENTS

CHAPTER PAGE

r. TWO ASPECTS OF SCIENCE - I

II. SCIENCE AND NATURE - l6

III. THE LAWS OF SCIENCE - 37

IV. THE DISCOVERY OF LAWS - 5$
V. THE EXPLANATION OF LAWS - - 77

VI. MEASUREMENT - - IOQ

VII. NUMERICAL LAWS AND THE USE OF MATHE-
MATICS IN SCIENCE - 135

VIII. THE APPLICATIONS OF SCIENCE - 158

INDEX ..... 185

WHAT IS SCIENCE?

CHAPTER I
THE TWO ASPECTS OF SCIENCE

F | AHERE are two forms or aspects of science.

First, science is a body of useful and practical JT

-*- knowledge and a method of obtaining it. It is
science of this form which played so large a part in the
destruction of war and, it is claimed, should play an
equally large part in the beneficent restoration of peace.
It can work for good or for evil. If practical science
made possible gas warfare, it was also the means of
countering its horrors. If it was largely responsible for
the evils of the industrial revolution, it has already cured
many of them by decreasing the expenditure of labour
and time that are necessary for the satisfaction of our
material needs. In its second form or aspect, science
has nothing to do with practical life and cannot affecfjS-
it, except in the most indirect manner, either for good or
for ill. Science of this form is a pure intellectual study.
It is akin to painting, sculpture, or literature rather than
to the technical arts. Its aim is to satisfy the needs of
the mind and not those of the body ; it appeals to nothing
but the disinterested curiosity of mankind.

The two forms, practical and pure science, are probably
familiar to everyone ; for the necessity for both of them
is often pressed on the public attention. There is some-
times opposition between their devotees. Students of
pure science denounce those who insist on its practical
value as base-minded materialists, blind to all the higher
issues of life ; in their turn they are denounced as

IS '.SCIENCE ?

academic and unpractical dreamers, ignorant of all the
real needs of the world. If the two forms of science were
really inconsistent with each other, both sides could
present a strong case. Few would deny that, in some
sense, intellectual interests are higher and more noble
than material interests ; for it is in the possession of
intellectual interests that we differ from the brutes.
Indeed, it may be urged that the only reason why men
should care for their material interests, or why they should
care to obtain anything but mere freedom from the pains
of cold and starvation, is that they may have the leisure
and the freedom from care necessary to cultivate their
minds. All but the most base must have respect, if not
sympathy, for those who prefer to live laborious days in
the pursuit of pure learning rather than to devote their
energies to the attainment of personal wealth and ease.
But to press this point of view is to misrepresent the
issue. More than the interests of the student himself
are involved ; and though the benefits of pure and
abstract science may be higher than those of practical
and useful science, they are much less widely distributed.
It is only a small minority of mankind who can hope to
share the former ; few have the mental equipment
necessary for the full enjoyment of the quest and dis-
covery of pure knowledge ; and of these few not all are
able to undertake the long and strenuous training thai
is a necessary preliminary to full enjoyment. On the
other hand the benefits of practical science might be
shared — even if they are not shared in our present
society — by almost every one ; the vast majority do not
possess the freedom from material cares necessary foi
the full development of their higher interests ; and il
practical science can so facilitate the satisfaction o*
material needs as greatly to increase the number who
have that freedom, its value, even if judged by the least
material and the most academic standards, may be in no

THE TWO ASPECTS OF SCIENCE 3

way inferior to that of the purest and most abstract
learning.

However, to-day it is probably unnecessary to pursue
such arguments. For it is now generally recognized that
the two forms of science, whatever may be their relative
value, are in fact inseparable. The practical man is
coming to understand that the earnest pursuit of pure
science is necessary to the development of its practical
utility, though he may sometimes have strange notions
of how that pursuit may best be encouraged. And
academic students are finding that the problems of practi-
cal science often offer the best incentive to the study of
pure science, and that knowledge need not be intellectually
uninteresting because it is commercially useful. In a
later chapter we shall consider in rather greater detail
what is the relation between pure and practical science
and why they are so inseparable ; but it is well to insist
at the outset upon their close connexion. For the
distinction between the two has undoubtedly discouraged
the study of science among the W.E.A. classes for which
this little book is intended primarily. Those who are
more familiar with the practical aspect are apt to think
that the study of science can be nothing but a disguise
for technical and vocational education ; while others
think that anything so entirely abstract as pure science
can have no bearing on the practical problems of society
in which they are more directly interested. Both views
are entirely mistaken ; the study of science need be no
more " technical " than the study of music, and, on the
other hand, it may be quite as practical as that of
political economy.

Nevertheless, though pure and practical science are
inseparable and merely different aspects of the same
study, it is necessary to remember the difference between
them. And I want to point out here, once and for all,
that what we are going to study directly is pure science ;

I WHAT IS SCIENCE?

that the motive of our study is supposed to be intellectual
curiosity without any ulterior end ; and that our criterion
will be always the satisfaction of our intellectual needs
and not the interests of practical life. This procedure
would be necessary even if our ultimate concern were
rather with practical science. For it is only if we under-
stand the nature of pure science that we can interpret
with confidence the knowledge that it offers and apply
it rightly to practical problems. Science, like everything
else, has its limitations ; there are problems, even
practical problems, on which science can offer no advice
whatever. One of the greatest hindrances to the proper
application of science to the needs of the community lies
in a failure to realize those limitations ; if science is
sometimes ignored, it is often because it has been dis-
credited by an attempt to extend it to regions far beyond
its legitimate province.

But it may be said, if the appreciation of pure science
must always be confined to a few serious students, what
is the use of such an attempt as this to make it intelligible
to the plain man ? The answer is simple : I only said
that the full appreciation must be so confined. Nobody
can appreciate good music to the full unless he has
trained himself by careful study, yet most of us can get
some value from a concert ; perhaps we get more actual
enjoyment than a skilled musician. It is just the same
with science. Indeed, there is little doubt that science
is the easiest of the branches of pure learning for the
amateur. It is quite common to find men of high
intellectual gifts and not without learning, who are
perfectly incapable of understanding what mathematics
or philosophy is all about, why anybody should ask suet
absurd questions and how they think anyone is the
better for the answers they give. A similar complete
indifference to science is not common ; almost every
one can be made to understand what science is

THE TWO ASPECTS OF SCIENCE 5

about, and almost every one derives some satisfaction
from the answers which it offers. This wider appeal
is often attributed to the practical interest of science,
but that explanation cannot be the whole truth ; for
some scientific doctrines, such as the Copernican theory
and the theory of evolution, have convulsed society
without having the smallest effect on anybody's
material comfort. The true reason is easy enough to
discover, but its complete discovery would answer most
of the questions which we are going to ask.

THE DEVELOPMENT OF PURE LEARNING

The main question which this book is designed to
answer may be expressed simply : What is Science ?
We have already answered it partly in saying that science
is a branch of pure learning which aims at intellectual
satisfaction. But it is not the only branch, and we must
ask next what it is that distinguishes science from other
branches. Is the distinction in the subject-matter that
it studies, or in the manner in which it studies it, or both
together, or, possibly, something quite different ? The
formal answer that I propose to give could be given at
once and quite briefly, but since at first sight it might
not appear plausible or even intelligible, we shall do
better to lead up to it more gradually.

All branches of pure learning spring from a common
stock. We generally think of V pure learning " as some-
thing peculiarly characteristic of the highest state of
civilization and as something which could develop only
when man had advanced a very long way from savagery.
But as a matter of fact the instinct which inspires pure
learning is one of the oldest and the most primitive ;
man begins to seek answers to the riddles which still
perplex the most abstruse of philosophers before he
{begins to wear clothes or to use metal implements.

6 WHAT IS SCIENCE?

Whether we regard the childhood of the race or of the
individual, we find that, as soon as man begins to think
at all, he utters his perpetual question, Why ? The world
around him does not appear to him immediately intel-
ligible ; it seems to have no meaning and to be arranged
on no comprehensible plan. He asks how the world
came to be what it is and why it is what it is. To such
questions, inspired in the first place by mere curiosity
rather than by a desire to control the world to his liking,
answers of some sort are given by the most elementary
religions and the crudest systems of magic. Some form
of religion or magic, which attempts to explain the world
in terms of ideas that are the product of thought and
reflection rather than of immediate perception, seems
characteristic of almost all races of men, however low their
intelligence and their material advancement.

It is, of course, impossible to determine certainly
whether these rudimentary attempts at pure knowledge,
which are found among the less developed races of to-day,
represent different stages in an evolution through which
all men's ideas have passed and must pass, or whether
they are entirely independent. And in particular it if.
impossible to trace back the history of our own puni
knowledge to its earliest origins. But we can trace it
back a very long way to the speculations of the ancient
Greeks in the third and fourth centuries before our era.
Greek thought, in the earliest stage in which we encounter
it, is very different from the primitive religions and magics
of savages ; but classical scholars find in it relics which
lead them to believe that its first origins were not very
different from the ideas of the most backward races o.
the present day. But in spite of these relics, the advance
that was made in the great Age of Greece was enormous.
It has largely determined all subsequent Europear
thought ; and it is not too much to say that there was
less advance made in pure learning in the 2,000 years

THE TWO ASPECTS OF SCIENCE 7

from 300 B.C. to A.D. 1700, than in the 200 years from
500 B.C. to 300 B.C. All speculation on the nature and
meaning of the world throughout the Roman Age of
civilization, through the Dark Ages and through the
Mediaeval Age, drew its inspiration directly from the
Greek philosophers, and especially from Aristotle ; it is
not until the Renaissance is well advanced that a new
stream enters from a wholly independent source. And
even to-day, when there is no school of thought which
maintains the Greek tradition in anything approaching
purity, its influence is still potent. Its effect upon
language is still most evident ; we cannot speak upon
any abstract subject, or express any general idea, without
using words which are either Greek or direct Latin
translations of Greek words. And since words are an
indispensable instrument of thought, in using Greek
words we are bound to be influenced to some extent by
Greek ideas.

Now Greek learning formed a single whole. To-day
we distinguish many branches of learning — mathematics,
science, philosophy, history, and so on. But this division
is quite modern ; Greek thought made hardly any dis-
tinction between them. (Perhaps an exception should
be made of history, and also of the study of languages ;
the Greeks did not study languages ; they knew none
but their own.) Even at the beginning of the nineteenth
century, all learning was called philosophy or (less
frequently) science, and a man was called a philosopher
even if he studied what we should now call mathematics
or science. Until well on in that century the universities
recognized only one form of study as a means to a degree,
and that form included a little of most of the forms recog-
nized, and sharply divided, at the present day. The
reason is not to be found simply in the smaller body of
knowledge at that time, so that one mind could grasp all
that was known : there was a real absence of distinction

8 WHAT IS SCIENCE?

between what are now regarded as different kinds of
knowledge. Our ancestors would have strenuously
denied that a great mathematician could be ignorant of
philosophy or a great philosopher ignorant of science.
One of the widest differences between modern and
ancient thought is the recognition that there are inde-
pendent systems of thought and independent bodies of
knowledge, and that errors in one branch are not
necessarily accompanied by errors in another.

SCIENCE AND OTHER STUDIES

Of course the branches into which pure learning has
separated have been changed greatly since, and in virtue
of, their separation. None has been more affected in
this manner than science ; the great development of
science of the last century is intimately connected with
its divorce from philosophy. And the changes are
so great that it is perhaps hardly right to regard the
science of to-day as the same thing as the science which
was not distinguished from other studies in Greek and
mediaeval thought. Nevertheless this discussion has not
been irrelevant ; for it reminds us that science, like a'l
other attempts to satisfy the curiosity of man, has its
ultimate roots in the simplest and most instinctive
speculations. It shows us also that, however distinct
from all other kinds of pure learning the science of to-da v7
may appear, the exact line of division and the exa( t
criterion are likely to be difficult to lay down ; a di—
tinction that was overlooked for 2,000 years is not likeiv
to be discoverable by a casual inspection. Again
suggests that, since the separation of science has takop
place in times so recent, one way to discover the dis-
tinction may be to inquire into its history of the word.

This history is quite simple. When it was recognizer
that the studies which now form part of science require u

THE TWO ASPECTS OF SCIENCE 9

a separate name, they were called " natural philosophy "
in distinction to " moral philosophy " ; and they were
also called " natural science " in distinction to " moral
science " ; for at that time " philosophy " and " science "
had practically the same meaning and were used inter-
changeably, although the former was the commoner.
All these expressions survive ; at the older universities
a professor of natural philosophy is indistinguishable
from a professor of physics or chemistry ; and " moral
science " is a common name for what is more usually
called philosophy. That " natural philosophy " has
become almost obsolete while " natural science " sur-
vives, is due partly to the inexplicable vagaries of language
which determine, apparently at random, which of two
synonyms is to die out ; but it is also partly due to the
fact that the older branches of learning from which the
students of science desired to separate themselves wrere
more often known as philosophy than as science. Again
the " natural " has been dropped, and only the " science "
retained, partly by mere abbreviation (just as " omnibus "
has been changed into " bus "), and partly because
students of science were by no means averse from hearing
their study called " science " without any qualification ;
for " science " is simply the Latin for " knowledge," and
the implication that all that is not science is not know-
ledge, naturally flattered their vanity. And it is impor-
tant to remember this history. For the older and
more general use of the word to mean pure knowledge
in general, or indeed any kind of knowledge, has not
vanished ; and we must be on our guard against imagin-
ing that everything to which the words " science " and
" scientific " are attached to-day have anything more to
do with natural science than with any other kind of
knowledge. When a journalist speaks of a " scientific
batsman " he merely means that he is skilful and does
not imply that he is learned in physics or astronomy.

10 WHAT IS SCIENCE?

Here no doubt the more general use is clearly distinguished
from the more special, but some misunderstandings about
the science that we are going to consider probably arise
from this double use of the word.

SCIENCE AND NATURE

But why were these special branches of learning called
" natural " ? Not because they were more natural, in
the conversational sense, than any other, or even in the
Shakespearean sense (which means idiotic) ; but because
they were regarded as being especially concerned with
nature. And what is meant by " nature," and hew
is science especially concerned with it ? The term
" nature " has never been used in a very precise sense
capable of accurate definition, but it seems generally
to be employed in contradistinction to man ; nature,
we may say roughly, is everything in the world that is
not human.^ Nature is regarded as the antagonist of
man, the obstacle which he has to overcome and the
enemy he has to fight, although he may sometimes turn
the enemy into a friend by judicious action. This id'ja
will be found, I think, to underlie most uses of the word.
It is true that sometimes, and more particularly in the
middle of the last century, man has been regarded as
part of nature ; for instance, one of Huxley's best-known
books is called " Man's Place in Nature " ; but the view
that man was part of nature was felt to be rather hetero-
dox and startling, an overthrowing of many preconceived
beliefs ; indeed, the phrase was used by Huxley large ly
in order to challenge accepted opinion.

Again, the opposition of nature and man is reflectod
in the terms used to distinguish the branches of pu *e
learning which were most clearly separated from scienc ?.
They were termed " moral " philosophy or science. No N
" morals," even in the very general sense attributed .o

THE TWO ASPECTS OF SCIENCE 11

the term when used in this connexion, are particularly
human. Common sense divides the world into three
great divisions — man, animals and plants (or living beings
other than man), and inanimate objects. To the third
division the idea of morals is clearly inapplicable, whether
it refers to all mental processes or more particularly to
right conduct ; and it is applicable only in a very limited
degree to the second ; the first is its proper province.
The distinction between natural and moral philosophy
suggests at once that the latter is concerned especially
with man and his ways ; the former with everything
that is foreign and external to man. F Nature means
practically the part of the world which man regards as
external to himself.^

Accordingly it is suggested that science should be
defined as that branch of pure learning which is concerned
with the properties of the external world of nature. Its
business is to find out accurately what those properties
are, to interpret them, and to make them intelligible to
man ; the intellectual satisfaction at which it aims
would be secured completely if this external world could
be reduced to order and be shown to be directed by
principles which are in harmony with our intellectual^
and moral desires. On the other hand, science will not,
on this view, be concerned with anything distinctively
human ; it will not consider human thoughts and actions,
ask what those thoughts and actions are, or examine and
criticize them. And this suggested definition of science-
would probably have been accepted very generally
at the time when science was first distinguished from
other branches of learning under the name of natural
philosophy. Nevertheless, there are difficulties in
accepting it. For, according to the view that has
been put forward, all pure learning arose ultimately
from man's desire to understand the world ; it was
his opposition to the external world of nature

12 WHAT IS SCIENCE?

that started his inquiry and his search for explana-
tion. If, then, it -is this external world which is the
special province of science, we should expect to find that
learning would become more distinctively scientific (in
the modern sense) as we trace it back through the ages,
and that branches, other than science, which are now
separated from the common stem, would appear at only
a relatively late stage in the growth. Actually, of course,
we find exactly the opposite ; what is now recognized
as science, as the study of nature and the external world,
is the youngest and not the oldest of the departments
of pure learning. Again, there are undoubtedly studies,
usually accepted as sciences, which specifically deal with
man and not with the external world which is contrasted
with him ; psychology and anthropology are examples ;
how are they consistent with the view that science is
characteristically non-human ? Lastly, it is generally
recognized to-day that science differs from other branches,
not only in the subject-matter that it studies, but also
in the manner in which it deals with this subject-mat ;er.
Even if we could define the subject-matter of science as
being the external world of nature, we should stil) be
left with the inquiry, which is really more interesting,
why the difference in the subject-matter involves so gieat
a difference in the attitude towards it.

SCIENCE OR SCIENCES ?

These difficulties show that we cannot obtain ;.he
answer that we require to our question, What is Science ?
by simply accepting the answer that might have Icen
given a hundred years ago. On the other hand, it is
indubitable that this answer is part of the truth. To
that inquiry we shall proceed in the next chapter, ,nd
with it shall start the serious part of our discussion.
But before we proceed, we shall do well to consider v ry

THE TWO ASPECTS OF SCIENCE 13

briefly one other matter which belongs properly to this
preliminary stage. Are we right to speak at all of
" science " ? Every one knows to-day that there is not
one science but many. Physics, chemistry, astronomy,
geology, zoology, botany, physiology, psychology, and so
on, although all called " sciences," seem to be branches
of knowledge almost as separate as any science is from
philosophy. A chemist may be as ignorant of botany
as a philosopher of mathematics. Can we say anything
that is true of all these sciences and is not equally true
of mathematics or philosophy ? Well, that is one of the
questions that we have to answer, and our answer will
be affirmative ; we shall lay down a criterion which
appears to distinguish all sciences from any other branch
of pure learning. But a word may be said here about
the relations of the different sciences.

The division between them corresponds in part to the
crude common-sense division of the external world of
nature. Thus we find some sciences (zoology, botany,
physiology) dealing with living beings and others (physics
and chemistry) with inanimate " matter." Further we
can distinguish sciences which deal with particular
objects from those which deal with the common sub-
stratum of objects. Thus geology deals with one parti-
cular object, the earth; and astronomy with other
particular objects, the stars ; zoology and botany
consider particular animals and plants. On the other
hand physics and chemistry deal with the substances
of which all particular material objects are composed ;
physiology with the functions common to all living
beings. So far the divisions between the sciences lie
along the lines that we should expect if science is the
study of the world of nature. But such divisions can
only be made very roughly. The province that is actually
regarded to-day as belonging to each science is very
largely the result of historical accident ; one line of

14 WHAT IS SCIENCE?

inquiry leads to another, and a new line of inquiry is
often assigned to the science that was the particular
study of the first investigator of that line, without
discussion whether the allocation can be justified on any
formal principle.

^Such considerations clearly justify the view that
science is a single whole and that the divisions between
its branches are largely conventional and devoid of
ulterior significance. But, though science may be really
one, its range and complexity to-day is so great that
the most learned of mankind cannot profess to a serious
knowledge of any but a very small part of it. And
therefore perhaps I ought to justify and explain my
temerity in writing of science in general. I should point
out that physics is the only science of which I profess
an expert's knowledge, and that the discussion is bound
to be directed from the standpoint of a student of that
science. But it is generally admitted that physics is in
some sense more fundamental than any other science,
and that the results of physics constitute, in some sense,
the starting point of other sciences. Why there should
be that relation is a matter for subsequent inquiry ; but
the admitted fact of the relation makes it certain that,
if we decide what is physics, what is its fundamental
subject-matter and its method of dealing with it, we
shall have gone a long way towards answering similar
questions which may be raised concerning any other
branch of science.

However, there is one question which should be roted
here. The examples of the various sciences that have
been given include none of the studies that lie on the
border line. Every one is prepared to grant that botany
and chemistry and physics are properly called sciences,
though there may be some doubt exactly what they Lave
in common ; but there are two studies of wide interest
the claims of which to be sciences are not universally

THE TWO ASPECTS OF SCIENCE 15

admitted. I refer to history and economics. The
judgment on these claims cannot, of course, be properly
passed until our inquiry into the characteristics of science
are further advanced ; but it will be convenient to antici-
pate some of our conclusions in order to dismiss the
matter. When he has read the two following chapters,
the reader should consider the question for himself.

Pthe view to which I incline is that history cannot be
usefully grouped with the characteristic sciences, and the
reason will appear at once in Chapter III. The main
concern of history is not with laws, but with particular
events. The decision concerning economics is more
difficult. A civilized community is part of " nature "
and there is no reason for thinking that such a com-
munity may not be subject to laws in the scientific sense.
But I have very grave doubt whether any economic
" laws" hitherto enunciated are laws in that sense ; and
the basis of my doubt will appear in the next chapter.
Economics might be, and some day may be, a science ;
but at present it is not. That is my opinion ; but as I
profess no special knowledge of economics, it may easily
be wrong. But I think it is certain that economics,
whether or no it is a science,^is so^different from those
that we are going to consider that it would be rash to
apply to it any of the conclusions that we shall reach.

CHAPTER II

\
SCIENCE ; AND NATURE

WHY DO WE BELIEVE IN AN EXTERNAL WORLE ?

HOW do we come to have any knowledge at all
of the external world of nature ? The answer
is obvious. We learn about the external world
through our senses, the senses of sight, hearing, and
touch, and, to a less degree, those of taste and smell.
Everything that we know about the external world
comes to us from this source ; if we could neither see,
hear, nor feel, we should know nothing of what was
going on round about us, we should not even know that
there was anything going on round about us ; we probably
should not even form the idea that there is such a thing
as the external world.

So much is clear and indubitable. But now we aave
to ask a much more difficult question, and one concerning
which there has been much more difference of opirion.
Why do we regard our senses as giving us knowledge of
the external world ? Every one agrees that if we "iave
any knowledge of such a world, it is derived from v hat
we see, hear, and feel, and not from any other sorrce ;
but it is quite possible to doubt that what we see, 'iear,
and feel, does really give us that knowledge, or thn r we
are right in interpreting the evidence which we d rive
from our senses in the manner in which we do habitually
interpret it. It is rather difficult for those who are
unfamiliar with the controversies that have raged round
this matter to grasp the position of those who express
such doubts ; it seems to us so obvious that wher. we
hear a noise or see an object we are perceiving somer ing
external to ourselves. And the difficulty of grasping

16

SCIENCE AND NATURE 17

the position is intensified because all our habitual language
is based on the assumption that there is an external
world which we perceive. For instance, when I want
to call attention to the sensation of sound, I can only say
that " I hear a noise " ; but the very form of the words
which have to be used to convey my meaning imply that
the " noise" is something different from the "I," who
hears it. Nevertheless it is necessary to try to under-
stand how such doubts can be put forward.

They are based on the fact that the experience of
seeing an object or hearing a sound is an event which
takes place in my mind ; it is a kind of thought — if we
use the word " thought " to mean anything that goes
on in my mind. That fact is expressed when it is said
that " I " hear the noise. Though the noise may be the
same, the fact that " I " hear it is different from the fact
that "you" hear it; the first fact is something that
happens in "my" mind, the second something that happens
in " your" mind. The noise, or the thing that causes the
noise, may be something in the world of nature, external
to both you and me ; but the hearing of the noise, which
is the fact on which you and I base the conclusion that
there is a noise or that there is an external object making
a noise, that hearing is not something external ; it is
something internal to you or to me, according as you or
I hear it. This view, that the perception of an external
object is something internal to the person who perceives
it, is as much part of the common-sense attitude towards
the matter as the view that the perception gives evidence
of an external object.

But now we may argue thus. It is agreed that the
perception of an external object is something internal to
the perceiver ; it is one of the thoughts, or the mental
events, of the perceiver. On the other hand we do not
regard all thoughts of a perceiver as giving evidence of
an external world ; there are also thoughts which are

A-

IS WHAT IS SCIENCE?

purely internal and totally unrelated to the external
world. Indeed, it is such thoughts which give rise to
the idea of a perceiver who perceives the external world.
For I regard all my perceptions as "my" perceptions,
because they are all connected together by thoughts of
other kinds. Thus I can remember my perceptions and
call them to mind ; I can think about them and compare
one with another ; I can judge that they are pleasant
or unpleasant and desire that some and not others should
recur. These thoughts about my perceptions I regard
as characteristically internal to me ; they are just the
things which make up " me " ; once more, it is these
thoughts about my perceptions which make me regard
them as " my " perceptions. It is very difficult to convey
these sentiments in words, just because, as has been said
already, all words assume these sentiments. But I hope
that any reader who considers the matter will agree that
the conception of an external world, which I perceive,
is founded as much on the idea that there are thoughts
which are wholly part of me, and have nothing to do
with the external world, as on the idea that there are
other thoughts which, though they are also part of me,
are intimately connected with the external world, and
inform me of that world.

If this view can be grasped, the basis of the douDts
that we are considering becomes clear. Some of my
thoughts I regard as wholly internal ; others, forming
the special class of sensations or perceptions recei/ed
through the organs of sense, I regard as rather pan of
the external world. Why, it may be asked relevantly,
do I make this distinction ? If it is necessary to regard
some of my thoughts as wholly internal and giving
evidence about me and not about the external wo. Id,
why do I not so regard all my thoughts ? If one c'ass
of my thoughts does not give any information about in
external world or even any evidence that there is su :h

SCIENCE AND NATURE 10

a world, why do I regard another class as giving such
information and evidence ? Is it not at least reasonable
to regard all thoughts in the same way, and to dispense
altogether with the recognition of an external world as
the cause of some of my thoughts ?

No serious school of thought has seriously maintained
the position indicated in these questions. Indeed, to
maintain it or to argue about it would be impossible, or
extremely foolish, for anyone who believed it. For —
we shall revert to this point in a moment — if there is no
reason for believing in an external world, there is no
reason in believing that there are other people with
whom to argue or against whom to maintain a position.
The view that has been based on the contention that
sensations are only thoughts, and therefore, like all other
thoughts, internal rather than external, is not that
sensations give no evidence at all for believing that there
is an external world, but only that the information which
we derive from our senses about the external world is
not so simple and direct as we often imagine, and conse-
quently, that our first impressions about the external
world may be very far from the truth. However, for
our purpose it is necessary to press the more extreme
view, and to ask why we distinguish so sharply between
sensations and other thoughts, and why we regard the
former and not the latter as giving evidence of, and
information about, an external world. In pressing the
view, I have, of course, no intention of maintaining that
our habitual distinction is not valid ; I only want to
elicit what is the difference between the two classes of
thoughts which makes it valid. Our question is, What
is the difference between the thoughts which we call
sensations, and connect with our organs of sense, and
the thoughts which we call memory, or reasoning, or
will ; and why does this difference lead us to refer the
first class, but not the second, to an external world ?

20 WHAT IS SCIENCE?

THE CHARACTERISTICS OF SENSE-PERCEPTIONS

_ There are two such differences. In the first place, our
sensations are much less under our control than are our
other thoughts ; in the second place, other people agree
with us in our sensations far more than they agree with

is in our other thoughts. That is, in brief, the answer
which I propose to give to the question ; it must now
be explained and expanded.

The first distinction is that our sensations are less
under our control than our thoughts. They are not
wholly beyond our control ; for, if I close my eyes, I
can refuse to see, and if I do not put out my hand, I can
often refuse to feel. But if I do look at an object it is
wholly beyond my control whether I see that it is red
or see that it is blue ; and if I put my hand into the fire
I cannot help feeling that it is hot and not cold. On
the other hand, thoughts, other than sensations, are not
wholly under control ; I cannot always remember what
I want to, and I cannot always keep my attention on
my work ; even my will is sometimes not under control,
and I may feel that there is a conflict within me. But,
though in this matter our sensations and our other
thoughts may differ in degree rather than in kind, it will
probably be recognized that there is this difference, and
that it is part of the reason why we feel that our sensa-
tions are intimately connected with something external
and do not take their origin wholly within ourselves.
For what is not under my control is not really pare of
me ; what I mean by " me " or " myself " is simply
what is under the control of my will ; my will is myself.
(Of course this is one of the statements which it is
impossible to express accurately in language which
assumes the position which is under discussion.) I
recognize this fact-^most clearly in those curious cases
when there is an internal conflict of will, and when " m y "

SCIENCE AND NATURE 21

will seems divided against itself ; I then speak of the
antagonistic wills as if they were those of two different
persons. If I act in a way which is contrary to my
normal will, I say that " I was not myself." This feeling
that " I " am practically indistinguishable from my will,
and that what is not subject to my will is not me, is
undoubtedly one of the main reasons for referring sensa-
tions, which are often wholly independent of my will,
to a foreign and external world. It has also, as we shall
see, a bearing on the second and more important difference
between sensations and other thoughts, to which we must
turn next.

This second difference is that other people agree with
me much more closely about sensations than they do
about any other kind of thought. The fact, expressed
in that manner, is extremely familiar. If I am in a room
when the electric light bulb bursts, not only I, but every-
one else in the room (unless some of them are blind or
deaf), hears the explosion and experiences the change
from light to darkness. On the other hand, apart from
sensations, we may all have been thinking about different
things, remembering different things, following different
trains of reasoning, and experiencing different desires.
This community of sensations, contrasted with the
particularity of other kinds of thoughts, leads naturally
to the view that the sensations are determined by some-
thing that is not me or you or anybody else in the room,
but is something external to us all ; while the other
thoughts, which we do not share, are parts of the
particular person, experiencing them. This simple ex-
perience is probably the main reason why we have come
to believe so firmly that there is an external world and
that our perceptions received by our senses give us
information about it.

For this is the test which we apply in practice when any
doubt arises if we are experiencing sensations which give

22 WHAT IS SCIENCE?

information about the external world. In general we
have not the slightest difficulty in distinguishing such
sensations from other thoughts and other mental events ;
but there are exceptions to the general rule. Thus when
we awake from a very vivid dream there is often a con-
siderable interval in which we are not sure whether our
dream experiences were real ; we have been experiencing
perceptions so very like the sensations which tell us about
the external world that, if we were left to judge solely
on the basis of their mental quality, we should probably
think they were sensations and gave us information
about the external world. Doubtless we have all had so
many dreams and are so well aware of the circumstances
in which they are likely to occur that a very brief reflec-
tion is usually sufficient to enable us to decide whether
we were dreaming or were really hearing or seeing some-
thing. Nevertheless doubtful cases do arise ; and, when
they do arise, what do we regard as a certain test to
decide the matter ? Surely the test is whether anyone
else has had the same experience. If we suddenly awake
imagining that we have heard a banging at the front
door, and if there is somebody else in the room who
shows no sign of having heard anything, we conclude
at once that we were dreaming. But, if somebody else
also heard the noise, we have no further doubt that c urs
was a real sensation.

In the same way, but less frequently, people are some-
times subject to hallucinations when awake. If some-
body tells us that he has seen a transparent old gentleman
clanking chains about the passages and carrying his
head under his arm, our disposition to believe thai he
has seen a ghost will doubtless depend largely upon our
attitude to the general question of the existence of
ghosts. But, whatever that attitude may be, our belief
would be enormously strengthened if we found -hat
other people present at the same time and place ad

SCIENCE AND NATURE 23

experienced the same sensations. In fact, a very little
reflection will show that our recognition of the possibility
of dreams and hallucinations is based almost entirely on
the fact that there are circumstances in which the sensa-
tions of one person may not be shared by others ; dreams
and hallucinations are simply mental experiences which,
though almost indistinguishable by the percipient from
sensations, are distinguished from sensations by being
peculiar to the percipient and in not being shared by
others. The community of sensations is our chief and
final test that experiences are true sensations such as
give information about the external world ; if we apply
other tests, it is only because this chief test is not available,
and any other tests we may apply are based on the results
which we are accustomed to obtain with this chief test.

OUR BELIEF IN OTHER PEOPLE

But now we must inquire a little more deeply and face
a difficulty. We believe in the external world because the
sensations of other people agree with our own. But
what reason have we to believe that there are other
people ? In our discussion hitherto we have spoken
of the world as divided into two parts, man and nature,
and we have regarded the external world as the same
thing as nature. But it is not really the same thing.
If I divide the world into man and nature, you are not
part of nature ; but if I divide the world into an external
and an internal part, you are part of the external part.
" You " are not " me " and " I " am not " you," you are
part of my external world and I am part of yours. Nature,
the part of the external world that is not man, is the same
thing as that part of the world which is external to all
men ; but it is not the same thing as my external world
or as your external world. Accordingly, if I am asking
what evidence there is for an external world, I must first

24 WHAT IS SCIENCE?

make up my mind whether you and other people are to
be regarded as part of it. ^If I do not regard you as part
of the external world, it is unreasonable to regard the
community of your sensations with mine as giving me
evidence of an external world. For that community
only gives me such evidence if you are external to me ;
if your sensations are internal to me, it is clear that the
fact that they agree with mine does not justify the
argument for the external world that we have just been
considering. On the other hand, if I know that you
are part of the external world, it is quite unnecessary
to examine your sensations and to inquire whether they
agree with my own in order to prove that there is an
external world ; for if there is not an external world,
you cannot be part of it. It seems that whichever
alternative I adopt, the argument for an external world
based on the community of your sensations and mine
breaks down. \ Either I must know already what the
argument professes to prove, or it provides no proofs.
Let us therefore examine rather more closely why we do
all actually believe that there are other people.

Our reason for believing that there are other people
appears to be of this kind. There is attached to me a
portion of the external world that I call my body. It
is part of the external world because I can perceive it
by my senses ; I can see my own hand, just as I can see
any other external object ; I can hear my own voice ;
and with my hand I can feel my own eye. On the other
hand, I regard it is peculiarly attached to me, and as
"my" body, because it is very intimately under the control
of my will. I can move my hand and I can close or open
my eyes by simply desiring to so do ; it is much less
affected by obstinacy in the face of my desires than the
remainder of the external world. Now I know that
certain changes in this external object which I call r.iy
body, changes which I can perceive through my senses,

SCIENCE AND NATURE 25

are intimately connected with certain purely internal
feelings. Thus, if I bring my hand too near a hot body,
I can see that it is snatched suddenly away ; and I know
that this sudden motion is accompanied by the purely
internal feeling of pain and also by certain muscular
feelings which are associated with movement of my
body. Now I perceive through my senses other parts
of the external world which appear very similar to my
body, and these objects undergo associated changes
very similar to those which take place in my body.
Thus I may see another object, very like my hand,
approach the same hot body ; and if I see that, I shall
see it snatched away again, exactly as I see my hand
snatched away. But this time I shall not experience
any feeling of heat or any feeling of muscular motion.

To explain these observations I imagine that, just as
there is intimately associated with my body a mind,
namely, my own mind, so there is intimately associated
with each of these other objects, so similar in appearance
and in behaviour, another mind ; I call these other
objects " other persons' bodies," and the minds which I
imagine to be associated with them I call " other persons'
minds " or simply " other persons/' I believe there
are other people because I see other bodies reacting in
the same way as my body ; and, if any reaction of
my body is accompanied by some event in my mind, I
suppose that the reactions of these other bodies are
accompanied by similar events in the minds of the other
people.

I do not propose to inquire whether this line of argu-
ment is justified (if anything so elementary and so funda-
mental to all thought can be called argument) or whether
it avoids the difficulty to which attention has been called.
The reader must inquire for himself whether he can put
the evidence for the existence of other people in a form
which is wholly convincing and is also such that it is

26 WHAT IS SCIENCE?

possible to base on the existence of other people an argu-
ment for and a criterion of the external world, without
lapsing into the fallacy of a circular argument which
assumes what it pretends to prove. As we shall see in a
moment, it is not relevant to our inquiry to decide whether
such arguments are justified or, indeed, whether it is
possible to produce any valid arguments for the existence
of other people and of the external world. All that I
am concerned with here is to draw attention to the ideas
which undoubtedly underlie our habitual and common-
sense distinction between the internal and the external
world, or between other people and ourselves, on the
one hand, and nature on the other. The ideas which
are Jmportant for our further inquiry are :

i. That the conception of " myself," on which is
founded the conception of all other people, is intimately
connected with the mental experiences which we call will
L^r^yolition. A person is something that wills ; volition
is the test of personality ; nothing is a person^ or has
personality (at least of the human type) unless it is
characterized by a will ; all exercise of will is inseparable
from the recognition of ^-person who exercises i£ ; .and
everything that is directly subject to the same wm is
part of the same person.

| 2? Our belief in the external world, or at least of .that

; part of it which is called " nature," is based on our

Lperceptions received through our sense organs. And we

believe that these perceptions inform us of the external

world, partly because they are independent of our wills,

but more because other people agree with us in those

sensations.

f "3. Our belief in other people is based on an analogy

between the behaviour of their bodies and the behaviour

. of .our own. If the actions of other bodies are similar

to those of our own, and if those actions in our bodie> are

accompanied by certain thoughts in our minds, thu we

SCIENCE AND NATURE 27

believe that there are similar thoughts in the minds
of the other persons whose bodies behave similarly to
our own.

A DEFINITION OF SCIENCE

This discussion was started by the suggestion that we
could answer our question, What is Science ? by saying
ihat science consists in the study of the external world
of nature. For reasons which have been given already,
and for others which will appear in due course, I propose
to reject that definition of science. In its place I propose
to put another, which could have been offered before,
but, if it had been offered before the discussion which has
just ended; it would hardly have been intelligible. This
definition is T Science is the study of those judgments
concerning which universal agreement can be obtained^

The connexion between this definition and the ideas
that we have been considering is obvious. It is the fact
that there are things concerning which universal agree-
ment can be obtained which gives rise to our belief in
an external world, and it is the judgments which are
universally agreed upon which are held to give us infor-
mation about that world. According to the definition
proposed, the things which science studies are very closely
allied to those which make up the external world of
nature. Indeed, it may seem at first sight, that we are
practically reverting to the definition of science as the
study of nature and that there is little difference except
in words between the definition which is proposed and
that which has been rejected.

But there are two very important differences. In the
first place the mere omission of such terms as " nature "
and the " external world " is important. For these terms
represent inferences from the judgments that we are
considering. Nature is not sensations or judgments

28 WHAT IS SCIENCE?

concerning which there is agreement ; it is something
which we infer from such sensations and judgments.
And this inference may be wrong; As has been said,
nobody maintains that it is entirely wrong, but it is very
strongly held in some quarters that some parts of the
inference usually made by common sense are wrong and
seriously misleading. If we call science the study of
nature, we are bound to admit that, if the common-sense
view about nature is largely mistaken, there must also
be a considerable element of doubt as to the real value
of the conclusions of science itself ; in other words,
science must to some extent be subordinated to philosophy,
in whose province lies the business of deciding the
worth of the popular conception of nature. Against such
subordination students of science have always protested :
and they can maintain their protest if they adopt the view
that science studies, not the external world, but merely
those judgments on which common sense, rightly or
wrongly, bases its belief in an external world. And it
may here be noted that among the difficulties that are
avoided by the definition are those to which reference
was made on p. 23.

But there is a much more important difference. It is
true that the popular belief in the external world is
founded primarily upon the fact of agreement about
sensations ; but, in deciding what part of our experience
is to be referred to that external world, common sense
does not adhere at all strictly to the criterion on which
that belief is ultimately based. .We do not ordinarily
refuse to regard as part of the external world everything
about which there is not universal agreement. A very
simple example will illustrate this point. A moment
ago a book fell from my table to the floor : I heard a sound
and, looking round, saw the book on the floor. Now I
had no hesitation in referring that experience to some-
thing happening in the external world ; but there vas

SCIENCE AND NATURE 29

not, and there cannot be, universal agreement about it
or indeed any agreement at all. For I am alone in the
room and nobody but myself has ever had, or can ever,
have, any share in that experience. Accordingly our
definition of science excludes that experience of mine
from the judgments which science studies, although to
common sense it was certainly an event in the external
world.

Such a simple example indicates at once how very
much stricter than the common-sense criterion of extern-
ality is the criterion which must be satisfied before any
experience is admitted by our definition as part of the
subject-matter which science studies. Science, as we
shall see, really does maintain the criterion strictly,
while common sense is always interpreting it very loosely.
I do not mean to assert that common sense is wrong to
apply a less strict criterion — that is a question which
lies far outside our province ; all that I mean is that any
experience which fails to satisfy the strict criterion of
universal agreement, though it may be quite as valuable
as experience which does satisfy it, does not form part
of the subject-matter of science, as we are considering it.
Here is the distinction between modern science and the1
vaguer forms of primitive learning out of which it grew.
When the possibility of applying the strict criterion of
universal agreement was realized, then, for the first time
in the history of thought, science became truly scientific
and separated itself from other studies. All the early
struggles of science for separate recognition, Bacon's
revolt against mediaeval learning and the nineteenth-
century struggle of the " rationalists " against the
domination of orthodox theology, can be interpreted, as
we shall see, as a demand for the acceptance of the strictly
applied criterion of universal agreement as the basis for
one of the branches of pure learning.

30 WHAT IS SCIENCE?

IS THERE UNIVERSAL AGREEMENT ?

But objections are probably crowding in upon the
reader's mind. The more he thinks about the matter,
the more impossible it will appear to him that truly and
perfectly universal agreement can be obtained about
anything. The scientific criterion, he will think, may
be an ideal, but surely even the purest and most abstract
science cannot really live up to it in a world of human
fallibility. Let us consider for the moment some of the
objections that will probably occur to him.

In the first place, he may say that it is notorious that
men of science differ among themselves, that they accuse
each other of being wrong, and that their discussions are
quite as acrimonious as those of their philosophical or
linguistic colleagues. This is quite true, but the answer
is simple. I do not say that all the propositions of
'science are universally accepted — nothing is further from
my meaning ; what I say is that the judgments which
science studies and on which its final propositions are
based are universally accepted. Difference of opinion
enters, not with the subject-matter, but with the conclu-
sions that are based on them.

In the second place, he may say that, if absolutely
universal agreement is necessary for the subject-matter
of science, a single cantankerous person who chose, out
of mere perversity, to deny what every one else accepted
could overthrow with one stroke the whole fabric of
science ; agreement would cease to be universal ! Now
this objection raises an important issue. How do we
judge what other people think, and how do we know
whether they do agree ? We have already discussed
this matter from the standpoint of common sense and
stated our conclusion on p. 25. But, here again, science,
though applying generally the same criterion as common
sense, insists on a much stricter and deeper application

SCIENCE AND NATURE 31

of it. We judge men's thoughts by their actions. In
common life we generally use for this purpose one par-
ticular form of action, namely, speech : if a man says
"I see a table " I conclude that the thoughts in his mind
are the same as those in my mind when I say " I see a
table." And men are generally so truthful that we do
not often need to examine further. But sometimes we
may suspect that a man is wilfully lying and that the
relation between his words and thoughts is not normal
(although it is again a relation of which we have some
experience in our own minds), and we can often detect
the lie by examining other actions of his. Thus, if he
says that he cannot see a table, we may not be able to
make him change his assertion ; but we may be able to
induce him to walk across the room, after having
distracted his attention from the matter, and then note
that he, like ourselves, walks round the table and does
not try to walk through it. Such tricks are familiar
enough in attempts to detect malingerers in medical
examinations. But what I want to point out here is
that the method can only be applied to detect lies about
a certain class of matters. If a man says that he does
not believe that 2 and 2 make 4, or holds that an object
can be both round and square, I do not see that we have
any way whatsoever to prove that he does not believe
what he says he believes. And the distinction is clear
between the matters in which lying and imposture can
be detected and those in which it can not. As we detect
imposture by examining a man's actions, it is only in
thoughts and beliefs that affect his actions that we. can
find out certainly what he thinks or believes. There may
be actually universal agreement on the proposition that
2 and 2 make 4, but in this case the objection that we are
considering is valid. A single denier could upset that
universal agreement, and we should have no way of
discounting his assertion and proving that the agreement

32 WHAT IS SCIENCE?

really is universal. Accordingly, in defining science as
the study of judgments concerning which universal
agreement can be obtained, [We are limiting science to
judgments which affect action and deliberately excluding
matters which, though they may actually be the subject
of universal agreement, do not affect actionj This con-
clusion is important, because it enables us to separate
science clearly from pure mathematics and logic ; but
space cannot be spared to pursue this line of thought
beyond a bare reference to it.

A man may also fail to join in the general agreement,
not because he is lying, because he is suffering from some
hallucination. This possibility was noticed before
(p. 22), and then we distinguished hallucinations from
true sensations by the fact of the agreement of others.
But now we are applying the test of agreement much more
strictly, and the mere fact that the man is under an
hallucination and does not agree with others is sufficient
to make the test fail. However, the difficulty can be
overcome in exactly the same way as that arising from
lying. We study all the man's actions, and we usually
find that, while some of them are consistent with his
assertion that he does not agree, others are inconsistent
with that assertion ; and those that are inconsistent are
those which we know, from our own internal experience,
to be less directly connected with consciousness and less
liable to aberration. Our test, once more, is always
whether the man acts on the whole as we should act if
we shared the thoughts which he professes. Curious
instances of this nature have occurred in actual science ;
there have been people who professed to be able to see
or to hear or to feel things which other men could not see
or hear or feel. But so far the difficulty has always been
removed by setting " traps," even if the honesty of the
man is beyond doubt, and showing that his actions in
general are not consistent with his professions.

SCIENCE AND NATURE 88

But I mention this matter for another reason. There
are persons under what we may call permanent hallu-
cinations : colour-blind people are an instance. There
are people who say that, to them, two objects, which to
normal people appear one as pink and the other as greenish-
blue, appear exactly the same colour ; and no traps
set for them will show any inconsistency in their judg-
ment. They will maintain their position when all their
interests lie in an ability to distinguish the colours. In
such cases universal agreement cannot be obtained. Are
the judgments to be excluded from the subject-matter of
science ? The answer is, Yes ; they are excluded.
And the fact that they are excluded is a support for the
definition of science which has been offered ; for there is
no doubt that they would have to be included if science
studied simply the properties of the external world.
Strange as it may appear to the uninitiated, colour,
judged by simple inspection, is not a scientific conception
at all, and it is not a scientific conception because
universal agreement cannot be obtained about it. The
procedure adopted is this. We find that normal people
regard the objects A, B, C, . . . as all pink and the
objects X, Y, Z, . . . as all blue ; colour-blind people,
on the other hand regard A, B, C, . . . X, Y, Z, as indis-
tinguishable in colour. But we find also that there is
some other property in which both normal and abnormal
people find that A, B, C, . . . agree and that X, Y, Z, . . .
agree, while in respect of this property both normal and
abnormal find that A, B, C, . . . differ from X, Y, Z, . . .
When we find that, we regard this new property as
the true and scientific test of colour ; for about this
property we can obtain universal agreement. And we
call some people abnormal, not merely because they fail
to agree with the majority, but because the}^ fail to make
a distinction where it is universally agreed that there is
a distinction,

34 WHAT IS SCIENCE?

To make this important matter clear, it may be well

to suggest a procedure which might be adopted. We

might make both the normal and the abnormal people

look at the objects through a red glass. Through the

glass, of course, everything will look the same colour to

both normal and abnormal, but different objects will

appear different shades of the same colour. The pink

objects A, B, C, . . . will all appear the same shade,

and so will the greenish objects X, Y, Z . . . ; but the

former will appear a lighter shade than the latter ; and

they will appear a lighter shade, not only to the normal

people, who see the difference of colour when the red

glass is not interposed, but also to the abnormal, who do

not see this difference. Here then universal agreement

has been attained ; every one agrees that through the

red glass the objects look different. Accordingly we

regard the appearance through the red glass as a better

basis for science than the appearance without the red

glass ; we say that scientifically the objects are different

in colour if they appear different through the red glass,

and we call one set of people normal and the other

abnormal because one set agree and the other do not

with the distinction based on this truly scientific criterion.

It is a very remarkable fact that, wherever we find

such abnormal people under permanent hallucinations

(and we find them in regard to all the senses), we can

always find another test which, in the manner just

described, enables us to restore universal agreement. It

is this fact, which could not be anticipated, which makes

science possible, and gives its great importance to the

test of universal agreement.

But perhaps the reader may doubt whether there are
really judgments about which every one agrees, if we are
allowed to include people with the most extremely
abnormal sensations, such as the totally blind or the
totally deaf. The doubt can only be removed by quoting

SCIENCE AND NATURE 35

an example which can easily be given. Every one who
has any sense-perceptions at all, and can come into any
contact with the external world, has the feeling that
events occur at different times and that some occur
before others. This is an example of a judgment concern-
ing which there appears to be the truest and most perfect
universal agreement. If one person A judges that an
event x occurs before an event y, then anyone else, how-
ever abnormal his sensations so long as he can experience
the events at all, will also judge that x occurs before
y ; he will never judge that x occurs after y. If the reader
considers this example, I think he will feel that in such a
judgment of the order in which events occur it is almost
inconceivable that there should be anything but the
most perfect universal agreement. Such judgment,
and such only, form the proper basis of science.

However our objector may make a last stand. He
may say that, though it is barely conceivable that there
should be disagreement about such a matter, barely
conceivable events do sometimes occur. It is just
possible that disagreement might arise where now there
is the most perfect agreement ; what would science
do then ? The question is unanswerable. It is quite
impossible to say what we should do if the world was
utterly different from what it is ; and it would be utterly
different from what it is if there were not judgments
concerning which universal agreement is obtainable.
It would be a world in which there was no "external
world/' For though, as has been urged, the general
agreement on which popular ideas about the external
world is based is not always as perfectly universal as is
demanded by the criterion set up by science, a deeper
inquiry than we can under take here would show that
common sense, just as much as science, does employ
conceptions which would be meaningless if, in the last
resort and in some cases, perfectly universal agreement

36 WHAT IS SCIENCE?

were not obtainable. That is the true answer to all the
objections that we have just been considering ; it has
been useful to consider them, because we have been
enabled thereby to bring to light some matters important
in the procedure of science ; but the answer to all objec-
tions based on the difficulty that might conceivably
be encountered in obtaining universal agreement is that
such agreement is actually obtained, and that all our
practical life and all our thought are based on tfce
admission that, in some matters but not in all, it is
actually obtained.

There is, however, an objection of another kind which
may yet be raised, but, since the discussion of it leads
us directly into more strictly scientific inquiry, it will be
well to leave it to open a new chapter.

CHAPTER III
THE LAWS OF SCIENCE

WHY DOES SCIENCE STUDY LAWS ?

/HT^HERE was quoted on p. 28 an example of an
experience which could not be the subject-matter

-*- of science, according to our definition, because
there could not be universal agreement about it. A
book fell on the floor when only one person was in a
position to observe the fall. Now it may be urged that
this example is typical, not of a small and peculiar class
of events occurring in the external world of nature and
perceived by the senses, but of all such events. No event
whatever has been observed by more than a very small
minority of mankind, even if we include only persons who
are all alive at the same time ; if we include — and our
definition suggests that we ought to include — all men,
past, present, and future, it is still more obvious that there
can be no event concerning which they can all agree ; for
there is no event which they can all perceive. Are we
then to take the view that no event whatever is the proper
subject-matter for science ? And, if we take that view,
what is there left in the external world which can properly
be such subject-matter ?

The answer is that we are to exclude every particular
event from the subject-matter of science. It is here that
science is distinguished from history ; Justory _studies
particular events, but science does not. "What then does
does science study ? {jSgience studies certain relations
between particular events?] It may be possible for every
one to observe two events each of a particular kind, and
to judge that there is some relation between those events ;
although the particular events of that kind which they

37

38 WHAT IS SCIENCE?

observe are different. Thus, in our example, it is im-
possible for every one to observe that a particular book
fell to the floor and made a noise on striking it ; JDUI^ j£js
/possible for every one to observe that, if a book is pushed ' \
/over the edge of the table, it will fall to the floor and/
will make a noise on striking it. Concerning that judg-
ment there can be universal agreement ; and that agree-
ment will not be upset, even if somebody has never
actually observed a book fall ; so long as he agrees when
at last he is placed in the necessary circumstances, that
a book will fall and that, when it falls, it will make a
noise, then universal agreement is secured.

If we could imagine ourselves without any experience
of the external world derived from our senses, we might
doubt whether there actually are such relations concerning
which universal agreement can be obtained ; we might
expect that it would be as impossible to find universal
relations between events as to find universal events.
But we all know from our experience that there are such
^relations and we know of what kind these relations are.
^They are of the kind that have just been indicated ; the
universal relations that we can state are between events
which are such that, if one event happens, then another
event happens."] Again, there might conceivably be other
relations betwran events of a different kind, yet of the
same universality ; actually there are not — at any rate
if we interpret the relation just stated correctly. There
is a certain class of relations between events for which
universal agreement can be obtained, which is thereby
distinguished from other classes for which it cannot be
obtained. Indeed, we might almost say that it is only
this class which can be the subject of universal agreement ;
for the necessity that all men, even if they live at different
times, should agree imposes limitations on the form of the
relation. But we need not inquire into this abstruse
matter ; /all that is necessary for our purpose i> to

THE LAWS OF SCIENCE 39

there are certain relations between events *r
concerning which all men can agree. -*J

Our_defimtion then limits science to the study of these
special relations between events. And this conclusion,
though the form in which it has been expressed and the
reasons alleged for it may be unfamiliar, is very well
known and widely recognized. For the relation of which
we have spoken is often called that of " cause and effect " ;
to say that, if a book falls off the table, it will make a
noise when it strikes the floor is much the same as to say
that the nojgfi^sjthe effect of the fall,_and the faJUhe^ause
of the noise. Again, assertions of cause and effect in
nature are often called " laws " or " laws of nature " ;
in fact, the assertion that a book, or any other object,
will fall if unsupported is one of the most familiar instances
that is often offered of one of the most widely known laws,
namely the law of gravitation. Accordingly, all that we
have said is that science studies cause and effect and that
it studies the laws of nature ; nothing can be more trite
than such a statement of the objects of science. Indeed,
I expect that some readers thought that a great deal of
unnecessary fuss was made in the previous chapter and
that all our difficulties about the relation between science
and nature would have vanished, if it had been said simply
that sci^ce^^judjed^jiot nature, but the laws of nature.

However here, as so oitenTthe popular view, tfiough it
contains a large measure of truth, is nol the whole truth.
The meaning popularly attached to " cause and effect "
and to " laws " is too loose and vague ; " cause and
effect," in the conversational sense, includes some rela-
tions which are not studied by science and excludes some
that are ; the assertions which. are popularly regarded as
laws are not invariably scientific laws, and there are many
scientific laws which are not popularly termed so. The
value of our definition is that it will enable us to give a
more precise meaning to these terms, and to show clearly

40 WHAT IS SCIENCE?

why and where scientific and popular usage differ.
Accordingly, in the rest of this chapter we shall examine
the matter more closely.

THE DEVELOPMENT OF LAWS

First, we may note that there is an apparent difference
between the popular conception of the part played by
laws in science and that laid down by our definition. It
AS probably usually thought that it is the aim and object
of science to discover laws, that laws are its final result.
But according to our view nothing can be admitted
to the domain of science at all unless it is a law.Jor it is
onljrthe relations expressed^ by laws that are capable of
universal agreement. Laws are the raw^naterial, not
the final product. There is nothing inconsistent in these
two statements, but the mode in which they are to be
reconciled is important. Laws are both the raw material
and the finished product. Science begins from laws, and
on them bases other laws.

To understand how this may be let us take an example
of a law ; that used already is not very suitable for the
purpose ; the following will serve better : A steel object
will rust if exposed to damp air. This is a law ; it states
that if one event happens another will follow ; although
it is the result of common observation, it would usually
be regarded as lying definitely within the province of
science. But now let us ask what we mean by a steel
object, or by " steel." We may say that steel is a hard,
shining, white substance, the hardness of which can be
altered by suitable tempering, and which is attracted by
a magnet. But, if we express what we mean by steel
in this way, we are in effect asserting another law. We
are saying that there is a substance which is both shining,
white, and capable of being tempered, and attracted by a
magnet ; and that, t/it is found to be white and capable
of tempering, then it will be magnetic. The very idea of

THE LAWS OF SCIENCE 41

" steel " implies that these properties are invariably
associated, and it is just these invariable associations,
whether of " properties " or of " events/' that are
expressed by laws. In the same way " rust " implies
another set of associated properties and another law ;
rust would not be rust, unless a certain colour were
associated with a powdery form and insolubility in water.

And we can proceed further and apply the same analysis
to the ideas that were employed in stating that the
properties of steel are invariably associated, or, in other
words, that there is such a thing as steel. For instance,
we spoke of a magnet. When we say that a body is' a
magnet, we are again asserting an invariable association
of properties ; the body will deflect a compass needle
and it will generate an electric current in a coil of wire
rotated rapidly in its neighbourhood. The statement
that there are magnets is a law asserting that these
properties are invariably associated. And so we could
go on finding that the things between which laws assert
invariable relations are themselves characterized by
other invariably associated properties.

This, then, is one of the ways in which laws may be
both the original subject-matter and the final result of
science. We find that certain events or certain properties,
A and B, are invariably associated ; the fact that they
are so associated enables us to define a kind of object,
or a kind of event, which may be the proper subject-
matter of science. If the object or the event consisted
of A and B without any invariable association between
them, it might be a .particular object or a particular
event, and might form an important part of the
popular conception of the external world, but it
would not be proper subject-matter for science. Thus
the man Napoleon and the battle of Waterloo are
an object and an event consisting of various properties
and events ; but these properties and events are not

42 WHAT IS SCIENCE?

invariably associated. We cannot, by placing ourselves in
such circumstances that we observe some of the proper-
ties (for instance short stature, black hair, and a sallow
complexion), make sure that we shall observe the other
properties of Napoleon. On the other hand, iron is a
kind of object suitable for the contemplation of science,
and not a particular object, because, if we place ourselves
in a position to observe some of the properties of iron, we
can always observe the other properties. Now, having
found an A and B invariably associated in this way, and
therefore defining a kind of object, we seek another set
of associated properties, C and D, which are again
connected by a law, and form another kind of object.
We now discover that the kind of object which consists
of A and B invariably associated is again invariably
associated with the kind of object which consists of C and
D invariably associated ; we can then state a new law,
asserting this invariable association of (AB) with (CD) ;
and this law marks a definite step forward in science.
If it is in such a way that science builds up new laws
from old it clearly becomes of great importance to decide
what are the most elementary laws on which all the others
are built. It is obvious that the analysis which we have
been noticing cannot be pushed backwards indefinitely.
We can show that in a law connecting X with Y, X is
the expression of a law between A and B, and Y of a law
between C and D ; we may possibly be able to show again
that A is the expression of yet another law connecting
some other terms, a and b. But, in the last resort, we must
come to terms, a and b, which are not resolvable into other
laws, and which, therefore, are not proper subject-matter
for science by themselves, but only when they occur in
the invariable association (ab). What, ther^are- the
terms at which we arrive at length " by i his analysis ?
What arc the irresolvable laws which must lie at the basis
of all science ?

THE LAWS OF SCIENCE 43

No more difficult question could be asked, and I cannot
pretend to answer it completely, even for that small
branch of science which is my special study. The reason
for the difficulty is interesting, and we must examine it.

Let us return to our first " law," namely that steel
will rust if exposed to damp air. I said that the use of
the word steel implied an invariable association of
properties which is asserted by the more elementary law :
There is such a thing as steel. But if we look at the
matter closely we shall see ^that this is not really a law.
For there are many kinds of steel ; the substances, all
of which the man in the street would call equally " steel,"
are divided by the fitter into mild steel, tool steel, high-
speed steel, and so on. And the scientific metallurgist
would go further than the fitter in sub-division ; he would
recognize many varieties of tool steel, with slightly
different chemical compositions and subjected to slightly
different heat-treatments, which might be all very much
the same thing for the purposes of the fitter. Tjjrtjf we

f st.ef>l,,_wft are in effect

saying that the_association of the properties of steel is
notinvariabTe, that therecan be many su&stances which,
thougTTlfiey "agree in some of their properties, differ in
others. Thus everything anybody would call steel con-
tains, according to the chemist, two elements, iron and
carbon ; but most steel contains some other element
as well as these two, and these other elements differ
from one steel to another ; one contains manganese,
another tungsten, and so on. It is not a law that every
substance which contains iron and carbon (and has cer-
tain physical properties of steel) contains manganese ;
for there are substances which agree in all these respects,
but differ in containing nickel in place of manganese,
and in certain other physical properties which are not
common to all steels.
There may seem to be an easy way out of the difficulty

ij WHAT IS SCIENCE?

raised by finding that there is really no such thing as
" steel." It has been implied that there are certain
properties common to all steels. If we make the word
steel mean anything which has these common properties,
whatever other properties it may have, then, since these
properties are invariably associated, the proposition that
there is steel (in this sense) will be a true law. But if
we examine 4he matter closely enough, we find that there
are not really any properties common to ail steels ; we
can find common properties only if we overlook distinc-
tions which are among the most important in science.
All steels, we may say, contain iron and carbon and all
are capable of being tempered. But they do not all con-
tain the same amount of iron and carbon, nor are their
tempering properties all the same ; and the variation
in the amount of carbon they contain is associated with
important variations of their tempering properties. As
we shall see — if, indeed, it is not obvious — one^of_Jhe
most important distinctions which science makes is
between objects or substances which have all the samej
property, but have it in different degrees ; the study of'
such distinctions is measurement^jmdmeasurement is
[ essential to science! The deeper wTm quire, ~tEe"less we
shall feel inclined to regard the statement, there is
steel, as a law, asserting invariable associations We
shall want to break this law up into many laws, one
corresponding to each of the different kinds of steel that
we can recognize by the most delicate investigation ;
when we have pushed these distinctions to the utmost
limit, then, and not till then, we shall have arrived at
laws stating truly invariable associations between the
various properties of these different kinds of steel.
r Here we meet with a process in the development of
/ science precisely contrary to that which we considered
| before. We were then considering the process by which
1 science, starting from a relatively small number of law.-.

THE LAWS OF SCIENCE 45

found relations between the objects of which they are
the laws, and so arrived at new and more complex laws.
In the second process, science takes these simple laws. I
analyses them, shows that they are not truly laws, and (
divides them up into a multitude of yet simpler laws.)
These two processes have been going on concurrently*
throughout the history of science ; in one science at one
time one of the processes will be predominant ; in another
science at another time, the other. But on the whole
the first process is the earlier in .time. Science started,
as weFhave seen, from" the ordinary everyday knowledge
of common sense. Common sense recognizes kinds of
objects and kinds of events, distinguished from particular
objects and particular events by the feature we have just
discussed ; they imply trie, assertion of a law. Thus all
" substances," iron, rust, water, air, wood, leather, and
so on, are such kinds of objects ; so again are the various
kinds of animals, horses, sparrows, flies, and so on. Simi-
larly common sense recognizes kinds of events, thunder
and wind, life and death, melting and freezing, and so
on ; all such general terms imply some invariable asso-
ciation and thus are, if the association is truly invariable,
proper matter for the study of science. And science in
its earlier stages assumed that the association was invari-
able, and on that assumption proceeded to build up laws
by the first process. It found that iron in damp air
produced rust ; that poison would cause death.

But, as soon as this process was well under way, the
second process of analysis began ; it was found that the
association stated by the laws implied by the recognition
of such objects was not truly invariable. This discovery
was a direct consequence of the first process. Thus,
until we have discovered that steel in general rusts, we are
not in a position to notice that there are some steels
which do not rust. When we have found that there are
certain substances, otherwise like steel, but differing

46 WHAT IS SCIENCE?

from other steels by being rustless, we are for the first
time in a position to divide steels into two classes, those
which do and those which do not rust, and so to analyse
the single law implied by the use of the term " steel "
into two laws, one implied by the term rusting steel, and
the other by the term rustless steel. And so, again, when
we have found that steel is attracted by a magnet, we are
first in the position to notice that different objects,
hitherto all called magnets, differ somewhat in their
power of attracting steel ; we can break up the single
law, There aie magnets, into a whole series of laws
asserting the properties of all the various magnets
which are distinguished by their different power of
attracting steel.

This is actually the history of scientific development,
so far as the discovery of laws is concerned. And now
we can see why it is so difficult to say what are the
fundamental and irresolvable laws on which science is
ultimately built. ^Science is always assuming, for the time
being, that certain: laws are irresolvaHe^ the law of steel,
for example, in the early stages of chemistry. But later
\ it resolves these laws, and uses for the purpose of the
I resolution laws which have been discovered on the
{assumption that they are irresolvable. At no stage is
iFdennitely and finally asserted that the limits of analysis
have been reached ; it is not asserted even in the most
advanced sciences of to-day ; it i£_always__iecQgnked
that^ a law which at jpresentjyDpearsTomplete may later
be shown to state an association which is not truly invari-
able^ Moreover, tfieTnlefmingling of the two processes
leads to the result that a law which is regarded as final
in one connexion is not regarded as final in another.
We use the law that there is steel to assert the law that
there are magnets, and at the same time use the law that
there are magnets to assert the law that there is steel !
If we attempted to describe science as a purely logical

THE LAWS OF SCIENCE 47

study in which propositions are deduced one from the
other in a direct line of descent from simple ultimate
assumptions to complex final conclusions, this double
role of laws, partly assumptions and partly conclusions,
would cause grave difficulty. All scientific arguments
would appear " circular," that is to say, they would
assume what they pretend to prove. But the result
that follows from our discussion is not that science is
fallacious*, because it does not adhere to the strict rules
of classical logic, but that those rules are not the only
means of arriving at important truths. And it is essen-
tial to notice this result ; for, since logic was the first
branch of pure learning to be reduced, to order and to be
brought to something like its present position, there has
been a tendency in discussions of other branches — and
especially in discussions of science — to assume that, if
they have any value and if they do really arrive at
truth, it can only be because they conform to logical
order and can be expressed by logical formulas. The
assumption is quite unjustifiable. ^Science is true, what-
p.ygr^pyrmfi may say : it has, for certain mimistj|japt
for all, the intellectual value which is the ultimate test
of truth. If a study can have this value and yet violate
the rules of logic, the conclusion to be drawn is that those
rules, and not science, are deficient. Nevertheless, while
it is important to insist that science is not necessarily
bound by logical formulas, it may be well to point out
that the difficulty which we have been noticing can be
overcome to some extent. The difficulty arises because
we have regarded all the different laws of science as differ-
ent propositions, some of which give rise to others. J^
would probably be more accurate to regard all the
so-called kwsjo^science as one single law which Isalways
being extended and refined ; and if we take "that view
there can be no question of deducing one law from
another ; the difficulty does not arise. Much might be

48 WHAT IS SCIENCE?

said in further explanation and extension of this attitude ;
Tmt space forbids a more lengthy discussion, and, with this
hint, the matter must be left.

It may be noted that in the actual practice of science
none of these difficulties and complexities arise. Every
science starts, as has been said, from the crude and vague
laws which have been elaborated as the result of that
continuous tradition of experience which is called common
sense. And, just because they are so intimately part
of common sense, there is usually no difficulty whatsoever
in obtaining for them the universal agreement which
makes them the proper subject-matter of science. It is
only when science gets to work and, instituting a much
deeper and more thorough inquiry than common sense
would ever institute, finds that the relations asserted by
the laws are not strictly invariable, that the question
of doubting the laws arises ; and the very inquiry which
suggests the doubts suggests also how the laws may be
amended so that once more, for the time being, universal
assent for them may be obtained. It is not actually
difficult to get people to agree that there are such things
as air and water ; the actual difficulty is rather to make
them see that what they call air and water are really
many different substances, all differing slightly by small
distinctions which have been overlooked. When we
study the history and development of any actual science
— and it must be remembered that this book is only
intended to be an introduction to such study — we do not
find actually that difficulties are continually raised by a
failure to obtain universal agreement ; though at a later
stage it is easy to see that the supposed laws of an earlier
stage were not true laws and that agreement could not
have been obtained for them, at any one stage the
distinction between the laws which are assumed as funda-
mental and those which are based on them is perfectly
clear and definite. The criterion of universal agreement

THE LAWS OF SCIENCE 49

is important because it gives a reason why we do actually
select for the study of science those portions of experience
which are actually selected ; but it is not the criterion
which we consciously apply. The conscious criterion
for the subject-matter of science is rather that it has been
regarded hitherto as connected together by a relation )

of invariable_association such as is asserted by a law. /

I

DO LAWS STATE CAUSES AND EFFECTS ?

So far we have only considered half of the problem of
the laws of science. We have expanded and made more
precise the conception of a law of nature, have considered
why such laws are of such supreme importance for science,
and have inquired how they can be at once its starting
point and its goal. A law, we have concluded, is the^
assertaijofjm^^ the events or

properties or otheFThings that it declares to be invariajiy
associated are themselves collections of ..other invariably
^^K^^BHngs. But we have not attempted to ask
furthefwliat is meant by " invariable association." We
noticed in passing at the outset that it was often thought
that laws were concerned characteristically with rela-
tions of cause and effect. A cause and its effect are invari-
ably associated. The view is therefore suggested that
by invariable association we mean simply the relation of
a cause to its effect. Is that what we mean ? This is
the other half of our problem and to it the rest of the
chapter must be devoted.

We must naturally start by asking ourselves what
exactly we mean (or what we should mean) by " cause
and effect." This is a matter on which there has been
much discussion ; but the idea which underlies most
frequently the use of the terms seems to be this. We
imagine that whenever an event B happens, it happens
only because it has been preceded by some other event

4

50 WHAT IS SCIENCE?

A ; and, on the other hand, if A happens, it is sure to be
followed in due course by B. When we can discover
such a relation between two events, we say that A is the
cause of B and B the effect of A. A single example will
suffice for illustration. If my ringer bleeds, it is because
I have cut it. The cutting, which necessarily precedes
the bleeding, is the cause ; the bleeding which necessarily
follows the cutting is the effect.

However, this simple and familiar notion, like so many
equally simple and familiar to a first glance, appears
rather more complex and intricate on further examination.
The many difficulties which might be and have been
raised to the acceptance of this simple view are not
strictly relevant to our present purpose, but a few of
them may be noted for the information of the reader
unaccustomed to philosophical discussion. The first
difficulty is that there are certainly pairs of events, A
( and ^^Jiej^a^WY^greceJing the other, which we do
\ not regard as cause and effect ; for instance, birth invari-
\ ably precedes death, and yet we should not accept readily
Mhe conclusion that birth is the cause of death. Again,
sometimes B, though always following A, also always
precedes another A ; day always follows night, but it
also always precedes night ; is day or night the cause,
or is there no relation of cause and effect involved ?
Once more, even when we are clear that there is a relation
of cause and effect involved, it is often difficult to say
precisely which, out of many alternations, is the cause.
Death, for instance, may be the effect of natural causes,
or of a hundred forms of accident or violence. We
know that it must always be the effect of one of them,
but we are so uncertain of which is the cause in each par-
ticular case that a special form of inquiry is thought
necessary. How is this uncertainty consistent with the
invariable sequence of effect after cause which seems
assumed by the use of those words ? Such difficulties

THE LAWS OF SCIENCE 51

as these undoubtedly suggest that by cause and effect
we mean something rather more abstruse and certainly
more obscure than the simple invariable sequence of one
event after another, which seems usually to be regarded
as constituting the causal relation.

But this is not what we have to consider. For those
who have seriously maintained that the business of laws is
to state relations of cause and effect have always regarded
such relations as consisting merely of invariable sequences.
It is possible that if the terms are used in this sense they
do not coincide exactly with common usage, but, if that
is so, it will only be one more of the innumerable examples
where science has diverted a term slightly from its sense
in popular discourse. What we have to ask is whether,
in discovering scientific laws, we are simply establishing
invariable sequences in which one event or set of events
follows after another.

It may be admitted without further discussion that

sjtatements of invariaBIe seguences. So much follows at
once-Frbm our previousHIscussion. For, though we have
spoken hitherto more vaguely of invariable association
rather than of invariable sequence, it is obvious that, if
there is such a thing as an invariable sequence, itis
onejorni aLIm^

the qualities which we concluded were necessary to make
a relation the proper subject-matter of science ; invariable
sequence is a relation concerning which universal agree-
ment might be obtained, just because it is invariable.
On the other hand, it seems certain that there are such
things as invariable sequences, for it is doubtless within
the province of science to predict future events, for
example the motions of the stars and the changes of the
weather ; and how could prediction from present to future
be possible, unless it were possible to discover sequences
of events which are invariable and which always recur ?

52 WHAT IS SCIENCE?

But it is much more doubtful whether it is only, or
even mainly, such sequences which are studied in the
establishment of laws. Indeed, some of the examples
of laws that have been quotecT already "seem to state
relations which are not sequences. For instance, we
spoke of the law of the association of the properties of
steel or of a magnet. But properties are not events which
follow each other. It is not necessary, in order to prove
that a substance is steel, always to observe that it is
attracted by a magnet before it is observed that it will
rust in damp air ; there is no time-relation of any kind
between the two properties. The properties of a single
substance, the invariable association of which is asserted
by the law of that substance, are something quite inde-
pendent of the times at which they are observed. They
differ completely in this matter from events which are
related to each other as cause and effect.

AodJ:here are scientific laws of anptherkind which are

not concerned with the invariablesequences that con-
stitute cause and effect, namely, numerical laws, of
which we shall have much to say laterT^TinpOrtant
examples of such laws are those which state that one
magnitude is proportional to another. For instance,
Ohm's Law states that the electric current through a
conductor is proportional to the electrical pressure
between its ends, so that if the pressure is doubled, the
current is doubled. Here, again, there is no time-relation
involved; the law "states something about numbers
and the size of the things that they represent ; there is
no idea of one thing being before or after another.

But, if there are so many and such important laws
which are obviously not concerned with cause and
effect, how did the idea ever arise that the establishment
of causes and effects was the sole or main purpose of
scientific laws ? In respect of the first example which
has been quoted, that of laws which state the properties

THE LAWS OF SCIENCE 53

of a substance, the answer is undoubtedly that it has
not been recognized sufficiently that such propositions
are laws. For they are not usually called laws. But
the fact that the name is not applied to them is largely
the result of history. As ^we rioted^ laws of_ this type
are among the results which science accepts in the
first instance from the experience of common sense,
although it subsequently refines them and may change
them almost beyond recognition. Knowledge is dignified
by the imposing name of law only when it has been
arrived at by Hftlifafiratft anrUxmscious investigation, and
not when, like Topsy, it simply "^growedT* But it is
more difficult to explain why numerical laws, to which
the name " law " is applied characteristically, have not
been recognized as providing instances to^show that
causejmd effect is jjoLthej>ri1y relation Wltk^MchJaws
are concerned .

I think the real reason is to be found in a confusion
between the method by which knowledge is attained
and the content of the knowledge once it is attained.
What I mean is this. Suppose we were seeking to
discover whether Ohm's Law is true. We shall set up
instruments for measuring the current and the pressure,
and shall then watch how the current changes when we
change the pressure. In . makmg juch^experiments, what
we shall actually observe^ that a change in current
follows a change in pressure ; we shall first make
deliberately a change in the pressure and then observe
a change in the current ; in other words, during the
experiment the change in current appears as an effect
of which the change of pressure is the cause. But,
though it may be maintained that it is by observing
such relations of cause and effect that we discover
the truth of Ohm's Law, it is not these relations which
are stated by the law. I It is the numerical relation, and
not the relation in time, thai is stated by the law. | For,

54 WHAT IS SCIENCE?

if we change our experimental arrangements a little, we
shall be able to alter the relation and interchange cause
and effect ; we shall be able first to alter the current
intentionally and then to observe a change in the pressure.
But, though we have thus turned cause into effect and
effect into cause, wre shall regard the experiments as
proving the truth of the same law, because the numerical
relation will be unaltered ; the same current will still be
associated with the same pressure. Thus, as was said
at the start, the law states a relation which is not that of
cause and effect, although it may be established by
observing such a relation ; there is a distinction between
the^meaning_Qi the law ano! tHe_^vib!ence~bn which it J^
assented.

'This distinction appears in experiments of all kinds
and is hardly separable from the fundamental idea of
an experiment. To make an experiment is practically
the same thing as to try what is the effect of some cause,
and in making it it is impossible not to think of the cause
before thinking of the effect. Thus, to revert to an earlier
example, if we are proposing to investigate what action
damp air has on steel, in order to make the trial we
must be thinking about damp air before we can know
what that action is. But when we discover that the
steel rusts, we see that the rusting is not the effect of the
damp air, in the sense that the presence of the damp
necessarily precedes the rusting ; we see that the rusting
is going on all the time that the steel is exposed to damp ;
the exposure and the rusting are concurrent, not con-
secutive.

A confusion between the order in which the processes
are present in our minds when we are making experi-
ments or observations and the relation which the pro-
cesses necessarily bear to each other — this confusion is,
I believe, the source of the notion, generally prevalent
during the last century, that cause and effect (in the

THE LAWS OF SCIENCE 55

sense of a pair of events occurring in an invariable
sequence) is a relation of peculiar significance to scientific
laws which are based on experiment. Its significance
for such laws is very much less than is generally believed.
n fact, it would hardly be too much to say that science
seeks to avoid entirely the necessity of recognizing such
causal relations, even when it is dealing with events
which actually do occur in invariable sequences^} Con-
sider, for example, a body falling to the ground. Each
position of the body invariably follows those higher up
and precedes those lower down. We might describe
the motion by saying that each higher position is the
cause of the lower positions and that the lower positions
are the effects of the higher. But actually we do not
adopt such a description. We regard the passage of
the body through the whole sequence of positions as a
single process which is not to be analysed at all ; it is
something which, as a whole, may have a cause (such
as the presence of the earth which attracts the body)
or an effect (such as the noise finally produced by its
impact), but in the process itself cause and effect are| *
not involved. This elimination of the causal relation, j (
and its replacement by the conception of a naturally 1 {
occurring process, is characteristic of all the more \
advanced sciences.

But, if the relation whirfi laws ^sfa.'h]^ between events
or properties or other things is not that of cause and
effect, what is it ? That is a -very interesting question,
but it is too difficult and needs too much detailed know-
ledge of science for any attempt to be made .here to
answer it. I think there are many slightly different
relations characteristic of laws ; the differences are
important and suggestive ; but they all agree in the
feature on which such stress has been laid already. They
may all be described as various forms of " invariable
association"""; and it is because they are all characterized

56 WHAT IS SCIENCE?

by this invariability that they are capable of being
experienced by everybody, and consequently are capable
of that universal assent which makes them the proper
subject-matter of science.

Nevertheless, there is one particular form of relation
involved jn laws which can be distinguished from others,
and on which emphasis may be laid once more. This
relation is that which characterizes^ what we have called
the law of the properties of a substance, or a kind of
system, the law, namely, which asserts that there is
such-and-such a substance or such-and-such a kind of
system, steel or magnets, for example. These laws, in
an elementary and imperfect form, are the earliest laws
of science, and they retain their peculiar significance
through much of its consequent development. The
recognition that there are such laws, and that they are
laws just as much as others more generally recognized
and called by that name, enables difficulties to be over-
come which have troubled some of those who have tried
to explain the nature of science. One of these difficulties
is connected with the " classificatory " sciences, such as
the older zoology and botany, or mineralogy. Such
sciences seem at first sight to state no laws at all ; they
simply describe the various animals, plants, or minerals,
and arrange them in groups according to their resem-
blances or differences, but do not state about them any
laws that are usually recognized as such. But, if such
sciences do not state laws, why are they regarded as
sciences ? We can answer now : They

laws of the kinfi which assorts tfre pi

system. In establishing that there is such an animal as a
cow and pointing out accurately its differences from a
sheep, or in investigating the differences between quartz
and rock-salt, zoology or mineralogy is discovering
laws, and laws of the kind that are of the most funda-
mental importance. The_ " ckssificatory " sciences differ

THE LAWS OF SCIENCE 57

from other sciences in that they confine themselves
to laws of this type and (in so far as they are completely
" classificatory ") do not base on them other laws of
other kinds.

To the student who, after reading this little book,
proceeds to the detailed study of one or more actual
sciences, the interesting problem is proposed of dis-
tinguishing between the different kinds of laws that are
characteristic of them ; for every science has its own
peculiarities in the features of its law. But we cannot
spend space on this inquiry ; we must now face a
different and more pressing problem.

CHAPTER IV
THE DISCOVERY OF LAWS

STATEMENT OF THE PROBLEM

FOR now, having decided what laws are and
what they state, we have to ask how they are
discovered. Laws state invariable associations ;
but how can we ever be "sure that an association
is invariable ? We may have observed an association
many times, and have always found that if one of the
associated events or properties occurs, the other occurs
also ; but if the association is truly invariable, we must
know, not only that the association always has been found
in the past, but also that it always will be found in the
future. Moreover, even if we have found the association
every time in the past that wre have looked for it, we
clearly cannot know that it has occurred when we have
not looked for it. The establishment that an association
is invariable and the assertion by a law that it is invariable
clearly require that we should be able to judge from the
observation of one or several occurrences of it all the
other occurrences that may happen or have happened.
How can we possibly attain such knowledge ?

One answer to this question is simply that we do not
know. We can never be certain that an event will
happen in the same way that we are certain that it has
happened. Indeed, there is a difference in the sense of
the word " know " applied to the two cases — a difference
in sense which is reflected by the use of different words
in most languages. When I have actually experienced
an event I have a direct and immediate perception of
it which is different in kind, and not merely in degree,
from my belief, however confident, that it will happen ;

58

THE DISCOVERY OF LAWS 59

it is not merely that I have more knowledge of it, but
that the knowledge is of a different kind. It is utterly
impossible that I should have of the one event the
kind of knowledge which I have of the other. If we
are to discuss profitably the problem before us, we must
remember this difference. We must not seek of events
which have not happened, the kind of knowledge appli-
cable only to those which have happened. And again,
we must not seek the kind of knowledge — it is once more
a different kind — that we have of purely logical or
internal propositions. When I say that a black cat
is black, I am quite certain that the statement is true
because by " a black cat " I mean a cat that is black ;
to say that a black cat is not black is not untrue ; it
is meaningless. The knowledge that I have of the
truth of the statement is necessarily different from that
which I can have of the statement that there is such a
thing as a black cat or that all cats are black ; and the
difference is once more in the kind of knowledge and
arises from a difference in the kind of statement ; it is
not a difference in degree of certainty.

The problem would be expressed better if we merely
compared our knowledge of various future events and
asked why we are more certain that some will happen
than that others happen and how we arrive at this
superior knowledge, for then we are sure of comparing
always knowledge of the same kind. Of some future
events we are as certain as we can be in respect of know-
ledge of this kind ; we are as certain as we can be that
the sun will rise to-morrow. It would be ridiculous to
say that we are not certain because we do not feel towards
that prediction the same mental attitude that we feel
towards the assertion that the sun rose to-day or the asser-
tion that to-day is not to-morrow. For, once more,
the difference in mental attitude necessarily arises
from the difference in the nature of the statements. All

60 WHAT IS SCIENCE?

that we can ask relevantly is why we are as certain that
the sun will rise to-morrow as we can be of any future
event and why we are so much less certain that it will
not rain to-morrow.

It is obvious that our certainty in one case and our
uncertainty in the other are derived from our previous
experience of the happening of similar events, and that
the differ encein knowledge is due to a difference m*tnTt
previous^expenence. Of course this statemenFdbes not
help us to~solve our problem, for since laws are undoubtedly
derived from previous experience, it is clear that it
is there that the foundation and evidence for them
must be found. But the form in which the problem has
been put enables us to avoid altogether a question to
which those who have discussed the matter have usually
devoted most of their attention. They have asked how
it is that previous experience gives any knowledge of
future experience and what justification there can be
for asserting in any case whatever that we have such
knowledge. The point of view that I tried to suggest
in the last few paragraphs is that this question is essen-
tially unanswerable because it is based on the neglect
of the fundamental distinction between different kinds
of knowledge. Our " knowledge " of future events
simply is something based on our knowledge of past
events ; when we say that we know something about
the future we only mean that we have a mental attitude
based on past experience ; and it is absurd to ask why
it is based on past experience, for, if it wrere not so based,
it would be something quite different. In my opinion
(though the reader should be warned that others would
dissent strongly) it can only lead to confusion of thought
to attempt to compare this knowledge with other kinds
of knowledge and to ask how they stand in relative
certainty. And yet some comparison of knowledge of
future events with other kinds of knowledge is always

THE DISCOVERY OF LAWS 61

intended when it is asked how we have such knowledge
from experience of the past.

And because such a question is meaningless the answers
given to it are meaningless also. They always consist
in some attempt to prove from very abstract and obscure
premisses a doctrine called by the high-sounding title of
Uniformity of Nature ; it is argued that, for some reason
to be found in transcendental philosophy, nature must
be such that what is true of her in one part, in one region
of space or at one period of time, must be true of her in
any other part. But the value of such a doctrine depends
entirely on the meaning attributed to " nature." If the
world means merely the non-human, external world of
common sense (as in Chapter II), then the doctrine is
simply untrue. Nature, in this sense, is not uniform ;
there are events which happen once and never happen
again ; and it is precisely because there are such events
that we distinguish between past and future. If it were
really true that " history repeats itself," there would be
no history ; history is the record of events which have
not repeated themselves and the proverb— like almost
all proverbs— merely represents an attempt to obtain,
by an epigrammatic form, credence for an assertion which
nobody would otherwise believe. It is true that many
events which do not so repeat themselves, and perhaps
the most important of these events, are characteristically
human and do not, therefore, form part of common-sense
" nature " ; but there are enough non-recurrent events,
which have nothing to do with man, to distinguish
between past and future and thus to controvert the
assertion that all nature is uniform in all its parts.

If, on the other hand, we mean by " nature " in this
connexion the carefully scrutinized nature of science,
then the doctrine merely states that nature is nature.
For this " nature " or external world of science is charac-
terized by and distinguished from everything else by the

62 WHAT IS SCIENCE?

fact that it is uniform ; for, as we have seen, it is made up of
invariable associations concerning which universal agree-
ment can be obtained. Any part of experience that is
not uniform would not consist of invariable associations
and would be at once excluded from this closely regulated
nature. ^Indeed the problem before us is simply that
of how we distinguish the uniform from the non-uniform
parts of the nature of common sense! for that is our task
in establishing the relations which are asserted by laws.
To attempt to base a method of making the distinction
on the assumption that all nature is uniform is simply to
misunderstand the problem that is to be solved.

AN ATTEMPTED SOLUTION

After this clearing of the ground, we can attack the
problem. What is the feature of our previous experience
which makes us so certain that the " law " of the rising
and the setting of the sun asserts a truly invariable asso-
ciation and, consequently, that the sun will rise to-
morrow ? In answer every one would say that our belief
is certain because we -have observed the association an
immense number of times without observing any failure.
And doubtless this is the reason in this particular case,
but other instances suggest that the answer is not funda-
mental or complete. For there are instances in which
an association which has been found as invariable has
at length been broken ; and there are instances in which
a law is asserted confidently as the result of a single
observation, so that there has been no chance of proving
any invariability. The instance of the first kind that is
always quoted in these discussions is that of the black
swan. Until Australia was discovered, swans had been
found invariably to be white in a very large number of
observations, and natural historians would have been
justified in asserting, according to the principle suggested,
the law that all swans are white ; and yet the law was

THE DISCOVERY OF LAWS 63

false, for some swans in Australia are black. Instances
of the second kind are plentiful in actual science. When
a chemist makes a new compound, he often determines
its melting-point or density ; as a result of a single
measurement he will often be prepared to assert that
its melting-point or density is higher (say) than that of
water, and nobody will dream of doubting that the asso-
ciation he asserts is invariable or that subsequent measure-
ments will lead to the same result.

These examples seem to prove that a^large number of
favourable instances, even if without exceptions, is nettjjiex:
suffioent_ jnorjwcessajy Jq^tablisl^ a, Jaw. But at the
same time they suggest what is the additional element re-
quired. V^e_haye_omitted to take into consideration other
laws^ closely .similar to those that are under discussion^
The chemist is certain that, in measuring the melting-
point of a new compound, he is establishing an invariable
relation, because from the examination of a great number
of other compounds he has found that the density is an
invariable property. On the other hand seventeenth-
century naturalists ought to have regarded with suspicion
a law that all swans are white (and probably they did
actually so regard it) because the examination of other
animals would have shown them that colour is by no
means an invariable property, but is liable to vary very
widely even among closely related species. In putting
the matter as we did, the full evidence was not disclosed.
The evidence for the invariable density of a new compound H
is not the single measurement of it, but the general law
that all densities are invariable properties. This law is
established by the observation, not of a single instance
or of one or two, but of a very large number of instances,
in none of which the relation has been found to fail.
The evidence for the assertion of the law of the density
of the new substance is really of exactly the same nature \
as that for the rising of the sun to-morrow.

64 WHAT IS SCIENCE?

This mode of expressing the matter is probably not
quite correct ; for closer examination would show that
it is difficult to regard the assertion that density (unlike
colour) is an invariable property as a true law. It would
be better to say that there are certain associations (such
as that of density or melting-point with the other pro-
perties of a substance) which, if they occur at all, we expect
to be invariable. In other words, we eyr^± ±r ^n(j jaws
of certain forms, and if we find an observation which
might be a particular instance of a law of one of these
forms, we are much more ready to jump at once to the
conclusion that this law is indeed true than we should
be if the law, of which the observation would be a
particular instance, is not of one of these forms. And
one of the reasons why we expect such laws is that
we have previously found a large number of them ;
however, as we shall see presently, this is not the only
reason.

THE ELASTICITY OF LAWS

But our answer is not complete yet. If this were all,
I think we should feel far more uncertainty than we
actually do feel about most of our laws. However many
favourable instances we had observed, if we felt that a
single unfavourable instance^if it occurred, would destroy
the law, we should never be free from uneasiness. The
contrary instance might occur ; we might go to our
laboratory one morning and find that the density of some
substance which we had measured the day before was now
quite different. Our confidence in the law is largely based
on the fact that such an unfortunate incident would not
necessarily destroy our belief in the law.

This statement may be surprising. Surely if a law
states that some relation is invariable ; and if, as we pro-
fessed in Chapter III., we are going to' be really strict in
our interpretation of invariability, then a single contrary

THE DISCOVERY OF LAWS 05

instance must destroy the law. For an. association which
has failed once, even if it has not failed a million times,
is not strictly invariable. True ; but what exactly is
the association we are asserting ? We are asserting that
a certain density is invariably associated with a certain
substance. If we find a new density we cannot maintain
the invariable association if we attribute it to the same
substance as that to which the old density was attributed.
But why should we not attribute it to a new substance ?
If we try the experiment over again and find that we do
not get the same result as before, what is to prevent us
avoiding any discrepancy between the two experiments
by simply saying that they are not made on the same
substance ?

Indeed this way out of the difficulty has been adopted
implicitly in the case of the black swan. Since we have
known of black swans, we do not say that there are not
white swans ; we recognize two kinds of swans, one of
which is black and the other white. Nor do we recognize
any error in the assertion, by those who did not know of
black swans, that all swans are white. All the swans
that they knew anything about were white and have
always remained white. The apparent difficulty arose
only because the new birds were called swans. If we
confine that term to the birds which were originally called
swans, any law about swans is quite unaffected by the
discovery of birds which resemble swans in some respects,
but which, since they are not wholly the same, should
not be called swans.

But, it may be urged, the case is not really parallel to
that which we must suppose if we want to face the
difficulty fairly. Black swans differ from white in other
things than their colour, so that there is a reason quite
apart from their unexpected colour for distinguishing
them from white swans. Again, even after the discovery
of black swans, white swans could still be found. But

66 WHAT IS SCIENCE?

suppose — we will return now to the instance of density —
that when we re-determined the density and found it
changed, we could not detect any change in any other
property of the substance and that we could not find
a substance which, resembling this substance in all other
respects, had the density found in the first experiment,
could we then maintain the invariability of the associa-
tion ? Well, it would doubtless be very awkward and
men of science would get into a fever of excitement, but
they could maintain their law. For the supposition that
nothing had changed between the two experiments is
impossible to realize ; the mere fact that a previous
experiment had been made and that the second experi-
ment had been made after the first is sufficient to make
some change between the two. Of course our usual
conception of a substance excludes the idea that such
changes — a mere repetition of a measurement or the mere
lapse of time — could change its properties and make it a
new substance ; we should have to alter our conception
of a substance. But that conception has been already
altered so greatly since it was taken over from common
sense that there would be no impossibility and no insuper-
able inconsistency in maintaining that, since wre made
the first experiment, the substance on which we made it
had vanished from our ken and been replaced by some
other substance, which might naturally enough have a
different density.

Indeed we should have to maintain something of the
kind, for, whatever we might do, the fact would remain
that we have observed two densities which cannot be those
of the same substance and cannot be asserted by the same
law. Either we must include the two observations
under different laws, or we must leave one (or both)
of them outside laws altogether. We adopt this last
alternative if we regard the first measurement simply
as a mistake ; a mistake is something that is excluded

THE DISCOVERY OF LAWS 67

necessarily from the subject-matter of science and to
which, therefore, a law can have no reference. It is
quite possible that, if such a case as we are imagining
actually occurred, we should adopt this course ; but
it must be remembered that we might adopt the other
and remove the discrepancy, not by rejecting the obser-
vation, but by stating two laws. Which alternative
we shall adopt depends on all the circumstances, and
here it is convenient to note why the observation of a
very large number of favourable instances is important
in the establishment of a law. If we have based a law
on a large number of instances, and subsequently find
other instances apparently discrepant, then, if, when
we choose between the alternatives just mentioned, we
reject the law, we place all these large number of obsep
vations outside the province of science. And this wtT*
are loath to do ; we want to reduce as much as possible
of our experience to order by means of laws, and the
rejection of the whole of our past experience as one
great " mistake," accords ill with that purpose. When
we have ordered a very large number of instances by
means of a law, we shall want to maintain that law at
all hazards ; and we shall be much more willing to
introduce other laws to Jnclude instances apparently
discrepant, and so to avoid the necessity of rejecting
tEe material on which the original law was based, than

— O

we should be if we have only ordered a very small number
of instances.

THE PURPOSE OF LAWS

It will be seen that in this discussion the question from
which it started has almost been left out of account.
We asked how we managed to establish laws, by the
examination of our past experience, which were true
also for future experience. The considerations that
have been put forward suggest that this problem is not

68 WHAT IS SCIENCE?

answered, but is hardly contemplated at all, in the
actual discovery of laws. When we are seeking laws,
we are only thinking about the experience that we have
actually had ; and the problem which we seek to solve
is one that has reference only to that experience. We
,seek to order the experience, to change it from a miscel-
laneous collection of apparently unconnected observa-
tions to a connected series of particular instances of
a few wide principles. These principles by means of
which, and in terms of which, weirder our past experience
are laws ; they state, as has been said so often before,
associations between events and properties which have
proved in our past experience to be invariable. It is
because the associations have proved invariable through-
out this experience that by means of them we can order
the experience as many particular' instances of a few
principles. When our experience is increased by the
addition of observations which were future but are now
past, we_seek oncejnore io orderjn the same manner our
experience ; but in this increased

volume all experience is of equal value, that which was
future is in no way different from that which was past,
for all is now past. It 'may happen that the order estab-
lished for the original experience is equally valid for that
which we now have ; the portion that is added can again
be regarded as particular instances of the laws which
were established as a result of the original experience.
And if that happens, we have no reason to change our
laws. But if that does not happen, if the laws estab-
lished for the original experience do not prove valid when
the volume of it is increased, then we have two
alternatives. We may either . reject altogether the
ad3edexperience ancF^miat it is noFproper~subject-

matter for science, or we may alter slightly (or radically)
our laws, so that they now order satisfactory both the
and the ng\v. If we adopt the second alternative,

THE DISCOVERY OF LAWS 69

the new laws propounded must still be such that they
order the old experience, and they must therefore present
some features of great similarity to the old laws. Which
of the two alternatives we shall adopt depends upon
which method leads to the most satisfactory ordering of
the complete experience. For this reason the first
alternative is never adopted if the second is available ;
for it means that we must leave unordered a portion of
experience which we thought could be ordered.

This is, I believe, the attitude that is actually adopted
by men of science in establishing laws. And if that is
so, the conception of prediction does not enter into explicit
consideration at all. Wejiojiot try to findjaws tha
will predict ; J^£_gjtiyj£y to find laws thatjvyill jpj^j
experience thatjvve have. It is possible to adopt that
attitude because, although we know that we shall have
future experience which has not been taken into considera-
tion, that future experience can never force us to abandon
the ground we have gained and to " disorder " the order
that has been established. Whatever the experience
may be, it will be possible either to order the increased
volume of experience, or else to reject altogether from the
subject-matter of science some portion of it, leaving only
the remainder to be ordered.

THE VALUE OF LAWS

But to the practical man that attitude will not seem
very satisfactory. It appears to deprive science of all
objective value. If scientific laws are true, only becaus'e
they can always be re-interpreted so that nothing can
prove them false, then science is merely a childish game
unworthy of the attention of any serious man. If, when
science asserts that the sun will rise to-morrow, it only
means that, if the sun does not rise, we propose to alter
somewhat our laws of the solar system, science is mere
trifling. What the plain man means by that assertion

70 WHAT IS SCIENCE?

is that the sun will rise and that the expectation of its
rising is a sound basis for the conduct of life ; he does
not mean something that can be made true or false just
as we please. It was all very well — I can hear an objector
say — to insist at the beginning of the chapter that we
can have no " knowledge " of future events ; it is undeni-
able that we have some kind of knowledge which we
habitually use in our practical life ; and if the only kind
of knowledge that science admits is a determination never
to be proved wrong, then we must seek elsewhere for the
information that undoubtedly is to be had.

Of course, I do not deny all this ; and now I shall
try and show how the two points of view may be recon-
ciled. Men of science, though they pay no direct atten-
tion to prediction, are not really indifferent to the success
of their predictions, interpreted in accordance with the
plainest common sense. If their predictions always
failed, it would mean that each addition to experience
would mean a new ordering of the whole. This ordering
doubtless could be accomplished in some fashion, but it
would have no value. The achievement of science would
be like that of Penelope, who wove a cloth that she
unravelled each night and started afresh each morning.
If all our predictions were failures, we could, I suppose,
continue our task of ordering experience, but no sane
r~man would do so. Science is only worth while because
I it does make real progress. The ordering established for
• L past experience is on the whole valid for future experience.
The exceptions are comparatively few, and, even when
they occur, it is found that the alteration of the order
is so slight — it is often only a natural development of the
old order — that the necessity for repeating the task is
not wearisome. Time unravels, not the whole web, but
only a few minor portions in which the shuttle has gone
awry. Scientific laws do predict exactly in the manner
which the plain man desires ; and it is really as necessary

THE DISCOVERY OF LAWS 71

for the purposes of science that they should do so as for
the purposes of practical life.

THE FUNDAMENTAL QUESTION

But why do they predict ? We return once again to
the question which we cannot avoid. The final answer
that I must give is that I do not know, that nobody knows,
and that probably nobody ever will know. The position
is simply this. We examine our past experience, and
order it in a way that appears to us most simple and satis-
factory ; we arrange it in a manner that is dictated by
nothing but our desire that the world may be intelligible.
And yet we find that, in general, we do not have to alter
the arrangement when new experience has to be included.
We arrange matters to our liking, and nature is so kind
as to recognize our arrangement, and to conform to it !
If anyone asks, Why, what kind of answer can we possibly
give ; how can we explain why the universe conforms to
our intellectual desires ?

Here we inevitably touch upon profound problems,
which lie far beyond the scope of this little book. I can
only say that, for myself, none of the answers that have
been offered seem satisfactory explanations, or even
explanations at all ; they raise more questions and more
difficult questions than they answer. But it may be
well to draw attention to two considerations that have to
be taken into account in any discussion of the matter.
The first has been mentioned several times before. It
must always be remembered that science does not
attempt to order all our experience ; some part of it,
and the part perhaps that is of most importance to us as
active and moral human beings, is omitted altogether
from the order. And it is very hard to say whether
we omit it because we know that we cannot order
it in the same manner as that which forms the

72 WHAT IS SCIENCE?

proper subject-matter of science, or because we feel
instinctively that, even if we could force it into such
an order, that order would not be appropriate to it.
I incline to the second alternative ; it seems to me
that there is something so fundamentally different
in the internal and external worlds (of Chapter II) that
we would not, even if we could, group them in the same
categories. But whichever alternative we adopt, it
remains equally difficult to explain why even the limited
part of experience which science takes as its province
conforms so closely to our desires, or why there should
be a part which can be selected so that it conforms.

The other consideration arises when it is asked who are
the " we," to whose intellectual desires nature conforms.
It is a grave difficulty, inherent in all the many attempts
to lay down rules whereby science may discover laws
valid for future experience, that they would indicate
that anybody who knew the rules could discover laws.
But that is not the fact ; it is not every one who has that
power. Indeed the fact seems to be precisely contrary.
Those who have professed the most intimate knowledge
of the rules, the great philosophers of science, such as
Bacon or Mill, have never been able to apply their rules
to the discovery of any law of the slightest value. Laws
have been discovered for the most part by people naively
innocent of all philosophical subtleties. The great man
of science, like the great poet or the great artist, is born
and not made ; like the artist he must train his faculties,
but training alone will not confer them. The vast maj ority
of mankind (a majority which includes a great many of
those who have done useful scientific work) cannot
discover laws, except in so far as they are helped, in a way
we shall notice immediately, by the previous work of an
infinitesimal minority. Either they cannot see order in
experience at all, or the order which they think they see
does not prove to be that to which nature is prepared to

THE DISCOVERY OF LAWS 73

conform ; they do not discover laws, or the laws that
they discover predict falsely. It is only the great leaders
of science who see the right order. They, and they only,
can establish an order which satisfies their intellectual
desires and yet find that it is valid for the future as well
as for the past. They, and they only, are in such har-
mony with the universe that it obeys the dictates of
their minds.

THE SIGNIFICANCE OF GENIUS

I fear this point of view will seem to some readers too
mystical for their tastes. Nevertheless I would press it
strongly on their attention. Of course I do not claim in
the least that it explains why laws, devised even by the
greatest of men, do predict, but it is necessary for the
understanding of science, as much as for the understand-
ing of art, to recognize that there are great men who
surpass their fellows in some scarcely comprehensible
manner. Science would not be what it is if there had
not been a Galileo, a Newton or a Lavoisier, any more
than music would be what it is if Bach, Beethoven and
Wagner had never lived. The world as^j&ejmow it^is
the prpjto^nts^enmses — and there may be evil as well
as beneficenT genius — and to deny that fact, is to stultify
all history, whether it be that of the intellectual or the
economic world.

But in one, as in the other, genius itself is too rare and
too short-lived to achieve much by its unaided efforts.
Great men — and this is particularly true of the greatest —
achieve more by their influence than by their direct action.
They change the world by enabling others to complete
what they have themselves begun. And in no direction
is this more true than in science. By far the greater
part of scientific work has been done, and by far the
greater number of laws discovered, by those of us who
have not the remotest claims to genius or any but the

74 WHAT IS SCIENCE?

very pedestrian talents of energy and application. But
we are simply following in the footsteps of our masters,/;1
In Chapter III we noticed that ther6~Xve?e~ standard
forms of laws ; there are many laws, all quite distinct and
ordering quite different groups of fact, which are yet
obviously all of the same form. The laws' asserting the
properties of a substance provide a notable example ;
there are many substances, but .the laws which assert
that there are such substances haye all the same form.
The properties of hydrogen, which are asserted by the
law that there is such a thing as hydrogen, are quite differ-
ent from the properties of iron, asserted by the law that
there is such a thing as iron ; yet the laws are of the same
form. Now once we have got the idea that there are laws
of this form, it is a comparatively simple problem, which
can be solved more or less according to fixed rules, to,
establish the laws of a new substance _or, bjf finding new
properties,_to^alter or augment an old one. ~And we
doTEnow now that laws of this particular form are among
those to which nature will conform, and which can be
usefully applied in prediction. The stroke of genius was
that of the man who first suggested a law of that form ;
once he had suggested it and showed that such a law is
permanently valid, it was easy enough for others to take
up the work and find others of the same form.

The discovery of the laws of substances is hidden in
the darkness of the past ; they are among the ideas which
we take over from common sense, and were invented by
the unknown giants who laid the basis of human know-
ledge. But advances quite as important, the discovery
of other forms of laws which have been used by the
humbler folk who do the spade-work of science, such
advances have occurred in historic times. Certain great
men are recognized as the founders of certain branches
of science, and if we inquire why they are so regarded,
we shall usually find (but another reason will be found in

THE DISCOVERY OF LAWS 75

the next chapter) that they were the first to establish a
law of the form that is specially characteristic of that
science. Thus of physics, numerical laws (which we shall
discuss later) are especially characteristic ; Galileo was
the first to establish a numerical law of the type of which
almost all modern physics consists ; nine-tenths of the
work of later physicists in the discovery of laws has been
simply the extension of laws of Galileo's form to other
fields of experience. Galileo may fairly be hailed as the
founder of experimental physics. Other great men have
so changed or amplified the form, that their work ranks
as independent — Boyle, and Ampere may claim place in
this class ; but again their fame rests largely on the dis-
covery of a new type of law which has been simply
applied elsewhere by lesser men. Of other sciences I am
not competent to speak, but if Lavoisier is the founder
of modern chemistry it is because he first established a
law of the form that asserts chemical combination ; and
if Linnaeus is the founder of systematic botany, it is
because he first established a law of the form that asserts
the existence of a particular species of plant.

This then is really the solution of the main question
of this chapter, as it faces the practising student of science.
He believes that if he cavaw of acertain

_

form'jnd order h^^enc^~® waY then that
law will predict and nature will conform to that order.
And so far, at least since the seventeenth century, Iris
expectation has never been falsified ; I believe that in
the history of modern science there is no instance of the
abandonment of a type of law which has once been firmly
established. Progress has been continuous ; it has con-
sisted in the establishment of many laws of old types, and
very occasionally, in the introduction of new types.
Even when at first sight experience has contradicted ex-
pectation, it has always been possible (as in the example of
the black swan) to remove the contradiction by resolving

76 WHAT IS SCIENCE?

a law of one of these types into several laws of the same
type, or by changing it to a law of another known type.
And what are these types ? To answer that question
would be to expound all science ; I want only to encour-
age the reader to study science for himself and to find
the types. But I have already indicated some of the
more important types, such as the law of a substance,
the law of a particular kind of animafandlthe numerical
law r"ancTiF EaslSeen urged thafall these laws have the
important common feature thatjthe things between which
the}' assert invariable associations are themselvesLJnter-
connected by other laws. Those who have previously
read inthe j>Mosopliy"of science will be surprised that
the causal law is omitted, but the reasons of the omission
were given in the previous chapter. In physics, at least,
it is not an important type, though it possibly may be
in other sciences, such as meteorology and medicine. And
by omitting the causal laws, we can omit also all reference
to the " Canons of Induction " which were supposed by
an earlier generation to provide the one and only means
for discovering scientific laws. They are futile, because
the problem which they profess to solve is not one which
has ever troubled any intelligent person. They tell us
how, when we know that an event is the cause or effect
of another, we may discover of which other event it is
the cause or the effect ; as a matter of fact, the crudest
common sense, applied in everyday life, serves for the
purpose. We might similarly draw up canons for dis-
covering of what substance a given property is a property,
when it is once known that it is a property of some sub-
stance ; but here again the rules would be so obvious as
not to be worth formulating. The problem of science is
not to discover examples of laws when once we know what
kind of law to look for ; it is to know for what kind of law
to look. And that problem, as we have insisted before,
is insoluble except by the genius which knows no rule.

CHAPTER V
THE EXPLANATION OF LAWS

THROUGHOUT the previous chapter I wrote as if
the ordering of nature which is effected by laws was
all that was necessary to satisfy our intellectual
desires and so to fulfil the purpose of science. But really
when we have discovered laws, we have fulfilled only part
of the purpose of science. Even if we were sure that all
possible laws had been found and that all the external
world of nature had been completely ordered, there would
still remain much to be done. We should want to explain
the laws.

"^Explanation in general is the expression of an assertion
in a more acceptable and satisfactory form?} Thus if
somebody speaks to us in a language we do not under-
stand, either a foreign language or the technical language
of some study or craft with which we are not familiar,
we may ask him to explain his statement. And we shall
receive the explanation for which we ask if he merely
alters the form of his statement, so as to express it in
terms with which we are familiar. The statement in
its new form is more acceptable and more satisfactory,
because now it evokes a definite response in our minds
which we describe by saying that we understand the
statement. Again we sometimes ask a man to explain
his conduct ; when we make such a demand we are
ignorant, or pretending to be ignorant, of the motives
which inspired his action. We shall feel that he has
offered a complete explanation if he can show that
his motives are such as habitually inspire our own
actions, or, in other words, that his motives are familiar
to us.

mr*T 77

V

78 WHAT IS SCIENCE?

But expressions, or the ideas contained in them, may
be more acceptable and more satisfactory, on grounds
other than their familiarity ; and all explanations do
not consist of a reduction of the less to the more familiar.
Indeed it would seem that the explanations which, in
the view of the man in the street, it is the business of
science to offer do not involve familiar ideas at all. Thus
we may expect our scientific acquaintances to explain
to us why our water-pipes burst during a frost or why
paint becomes dirty sooner in a room lit by gas than in
one lit by electricity. We shall be told in reply that the
bursting of the pipes is due to the expansion of water
when it is converted into ice, and the blackening of the
paint to the combination of the white pigment with
sulphur present in coal-gas to form compounds that are
dark, and not white, in colour. Now in these instances,
the ideas involved in the explanation are probably less,
and not more, familiar than those that they are used to
explain. Many more people know that water-pipes burst
during a frost than know that water (unlike most liquids)
expands when it freezes ; and many more know that
their paint goes black than know that lead carbonate
(one of the commonest white pigments) is converted by
sulphur into black lead sulphide.

Why then do we regard our questions as answered ?
Why do we feel that, when we have received them, the
matter is better understood, and our ideas on it clearer
and more satisfactory ? The reason is that the events
and changes have been explained by being shown to be
particular examples of a general law. Water always
expands when it freezes, although it does not always burst
household pipes ; for it may not be contained in pipes or
in any closed vessel. And lead carbonate always reacts
with sulphur in the form present in coal-gas, even if it is
not being used as a pigment. We feel that our experience
is no longer peculiar and mysterious ; it is only one

THE EXPLANATION OF LAWS 79

instance of general and fundamental principles. It is
one of the profoundest instincts of our intellectual nature
to regard the more general principle as the more ultimately
acceptable and satisfactory ; it is this instinct which led j
men first to the studies that have developed into science.
In fact, what was called in the last chapter the
" ordering " of experience by means of laws might equally
well have been called the explanation of that experience,
f Laws explain our experience because they order it by
L&elemng particular instances to general principles] the
explanation will be the more satisfactory the more general
the principle, and the greater the number of particular
instances that can be referred to it. Thus, we shall feel
that the bursting of the pipes is explained more satis-
factorily when it is pointed out that the expansion of
water when converted into ice explains also other
common experience, for instance that a layer of ice
forms first on the top of a pond and not on the bottom.

Doubtless there are other kinds of explanation ; but
it is important for our purpose to notice that the explana-
tions of common life often depend on these two principles *
— that ideas are more satisfactory when they are more j
familiar and also when they are more general ; and that -
either of these principles may be made the basis of an
explanation.

When it is asked what is the nature of the scientific
explanation of laws — and it is the purpose of this chapter
to answer that question — it is usually replied that it is
of the second kind, and that laws are explained by
being shown to be particular examples of more general
laws. On this view the explanation of laws is merely an
extension of the process involved in the formulation of
laws ; it is simply a progress from the less to the more
general. At some stage, of course, the process must
stop ; ultimately laws so general will be reached that
for the time being at least, they cannot be included under

80 WHAT IS SCIENCE?

any more general laws. If it were found possible to
include all scientific laws as particular instances of one
extremely general and universal law, then, according
to this opinion, the purpose of science would be com-
pletely achieved.

I dissent altogether from this opinion ; I think it leads
to a neglect of the most important part of science and to
a complete failure to understand its aims and develop-
ment. I do not believe that laws can ever be explained
by inclusion in more general laws ; and I hold that, even
it were possible so to explain them, the explanation would
not be that which science, developing the tendencies
of common sense, demands.

The first point is rather abstruse and will be dismissed
briefly. It certainly seems at first sight that some laws
can be expressed as particular instances of more general
laws. Thus the law (stating one of the properties of
hydrogen) that hydrogen expands when heated seems
to be a particular instance, of the more general law,
that all gases expand when heated. I think this appear-
ance is merely due to a failure to state the laws quite
fully and accurately, and that if we were forced to state
with the utmost precision what we mean to assert by a law,
we should find that one of the laws was not a particular
case of the other. However, I do not wish to press this
contention, for it will probably be agreed that, even if
we have here a reference of a less general, to a more
general law, we have no explanation. To say that all
gases expand when heated is not to explain why hydrogen
expands when heated ; it merely leads us to ask immedi-
ately why all gases expand. An explanation which leads
immediately to another question of the same kind is no
explanation at all.

THE EXPLANATION OF LAWS 81

WHAT IS A THEORY ?

How then does science explain la ws_ ? It explains
them by means of " theories, n whicrT are not laws,
although closely relafe^iEoTaws. We will proceed at
once to learn what a theory is, and how it explains laws. 1
For this purpose an example is necessary, even though
its use involves entering more into the details of science
than is our usual practice. A great many laws are known,
concerning the physical properties of all gases ; air,
coal-gas, hydrogen and other gases, differ in their chemical
properties, but resemble each • other in obeying these
laws. Two of these laws state how the pressure, exerted
by a given quantity of gas on its containing vessel, varies
with the volume of the vessel, and with the temperature
of the gas. Boyle's Law states that the pressure is
inversely proportional to the volume, so that if the volume
is halved the pressure is doubled ; Gay-Lussac's states
that, at a constant volume, the pressure increases propor-
tionally to the temperature (if a certain scale of tempera-
ture is adopted, slightly different from that in common
use). Other laws state the relation between the pressure
of the gas and its power of conducting heat and so on.
All these laws are " explained " by a doctrine known as
the Dynamical Theory of Gases, which was proposed early
in the last century and is accepted universally to-day.
According to this theory, a gas consists of an immense
number of very small particles, called molecules, flying
about in all directions, colliding with each other and with
the wall of the containing vessel ; the speed of the flight
of these molecules increases with the temperature ; their

1 The reader should be warned that he must remember that the
word " theory " is here used in a strictly technical sense of which the
meaning is about to be explained. He must not attach to it any of the
ideas associated with the word in ordinary language. In Chaptei VIII
some reference will be made to the use of " theory " in centra-distinction
to " practice."

82 \V1-IAT IS SCIENCE?

impacts on the walls of the vessel tends to force the walls
outwards and represent the pressure on them ; and by
their motion, heat is conveyed from one part of the gas
to another in the manner called conduction.

When it is said that this theory explains the laws of
gases, two things are meant. The first is that if we
assume the theory to be true we can prove that the laws
that are to be explained are true. The molecules are
supposed to be similar to rigid particles, such as marbles
or grains of sand ; we know from the general laws of
dynamics (the science which studies how bodies move
under forces), what will be the effect on the motions of
the particles of their collisions with each other and with
the walls ; and we know from the same laws how great
will be the pressure exerted on the walls of the vessel
by the impacts of a given number of particles of given
mass moving with given speed. We can show that par-
ticles such as are imagined by the theory, moving with
the speed attributed to them, would exert the pressure
that the gas actually exerts, and that this pressure
would vary with the volume of the vessel and with the
temperature in the manner described in Boyle's and Gay-
Lussac's Laws. In other words, from the theory we can
deduce the laws.

This is certainty one thing which we mean when we
say that the theory explains the laws ; if jthe laws
not be deduced from the theory, the thegi^woul
eXpIairTtheTaws^arrd the theory would not be true- But
tKrs"canfTOlrbe aH that we mean. For, if it were, clearly
any other theory from which the laws could be deduced,
would be equally an explanation and would be equally
true. But there are an indefinite number of " theories "
from which the laws could be deduced ; it is a mere
logical exercise to find one set of propositions from which
another set will follow ; and anyone could invent in a few
hours twenty such theories. For instance, that the two

THE EXPLANATION OF LAWS 83

propositions (i) that the pressure of a gas increases as the
temperature increases (2) that it increases as the volume
decreases, can be deduced from the single proposition
that the pressure increases with increase of temperature
and decrease of volume. But of course the single pro-
position does not explain the two others ; it merely states
them in other words. But that is just what logical deduc-
tion consists of ; to deduce a conclusion from premisses
is simply to state the premisses in different words, though
the words are sometimes so different as to give quite a
different impression. l If all that we required of a theory
was that laws could be deduced from it, there would be
no difference between a theory which merely expressed
the laws in different words without adding anything
significant and a theory which, like the example we are
considering, does undoubtedly add something signifi-
cant.

It is clear then that when we say the theory explains
the laws we mean something additional to this mere
logical deduction ; the deduction is necessary to the
truth of the theory, but it is not sufficient. What else do
we require ? I think the best answer we can give is that,
injorder that j^Jj^gp^ mav exglajs, we re<liure it — to
explainl We7equirel:liiar^ to ourjdgaSj and

that the idea¥ wTucir'it adds' sEall be acceptable. The
reader will probably feel that this is true of the explana-
tion of the properties of gases offered by the dynamical
theory. Even if he did not know (and he probably does
not know apart from what I have just told him) that the
laws can be deduced from the theory, he would feel that
the mere introduction of moving particles and the sugges-
tion that the properties of a gas can be represented as
due to their motion would afford some explanation of
those properties. They would afford some explanation,

1 The reader should be warned that some people would dissent
vehemently from this assertion,

81 WHAT IS SCIENCE?

even if the laws could not be deduced correctly ; they
would then offer an explanation, although the explanation
would not be true.

And this is, I believe, the reason why he would feel thus.
Only those who have practised experimental physics,
know anything by actual experience about the laws of
gases ; they are not things which force themselves on
our attention in common life, and even those who are most
familiar with them never think of them out of working
hours. On the other hand, the behaviour of moving
solid bodies is familiar to every one ; every one knows
roughly what will happen when such bodies collide with
each other or with a solid wall, though they may not
know the exact dynamical laws involved in such reactions.
In all our common life we are continually encountering
moving bodies, and noticing their reactions ; indeed,
if the reader thinks about it, he will realize that whenever
we* do anything which affects the external world, or when-
ever we are passively affected by it, a moving body is
somehow involved in the transaction. Movement is
just the most familiar thing in the world ; it is through
motion that everything and anything happens. And so
by tracing a relation between the unfamiliar changes
which gases undergo when their temperature or volume
is altered, and the extremely familiar changes which
accompany the motions and mutual reactions of solid
bodies, we are rendering the former more intelligible ;
we are explaining them.

That is to say, the explanation of laws offered by
theories (for this example has been offered as -typical)
is characteristically explanation of the first of the two
kinds with which the chapter started. It is explana-
tion by greater familiarity, essentially similar to that
offered when a statement is translated from an unknown
to a known language. This conclusion may be surprising,
and indeed it is not that generally advanced. Before

THE EXPLANATION OF LAWS 85

developing our view further, it will be well to examine
the matter from another point of view.

DIFFERENCE BETWEEN THEORIES AND LAWS

It was stated before that it has been usually held that
the explanation of laws consists in showing that they are
particular examples of more general laws. If this view
were applied to the example under discussion, it might
be urged that the dynamical theory explains the properties
of gases because it shows that they are particular
examples of the laws of dynamics ; the properties of
gases are explained because they are shown to be the
consequences of the subjection of the molecules, of which
the gases consist, to the general laws of all moving bodies.
Here, it might be said, is the clearest possible instance of
explanation by generalization, a simple extension of the
process involved in the discovery of laws.

But, against this view, it must be pointed out that the
most important feature of the theory is not that it states
that molecules are subject to dynamical laws, but that :
which states that there are such things as molecules, and j
that gases are made up of them. It is that feature of the /
theory which makes it a real explanation. Now this
part of the theory is not a particular instance of any more
general law ; indeed it is not a law or anything that could
be an instance of a law. For it is not, according to the
criterion laid down in Chapter II, part of the proper
subject-matter on which science builds its foundations.
Molecules are not things which we can see or feel ; they
are not, like the ordinary material bodies to which the
laws of dynamics are known to apply, objects discernible
to direct perception. We only know that they exist
by inference ; what we actually observe are gases, vary-
ing in temperature and pressure ; and it is only by these
variations that we are led to suspect the existence of the

80 WHAT IS SCIENCE?

molecules. We may apply once more our fundamental
test of universal agreement which serves to distinguish
the objects concerned in laws from any others. If some-
body denied the existence of molecules, how could we
prove him wrong ? We cannot show him the molecules ;
we can only show him the gases and expound the theory ;
if he denied that the theory proved the existence of the
molecules, we should be powerless. We cannot prove
by his actions that he is perverse or deluded ; for his
actions will be affected only by the properties of gases,
which are actually observed, and not by the theory intro-
duced to explain them. Actually the dynamical theory of
gases has been denied by men of science of high distinc-
tion. Usually the denial was based partly on the asser-
tion that the laws of gases could be deduced accurately
from the theory, but it has often been accompanied by
the contention that, even if they could be deduced
accurately, the theory was not true, and not worthy of
acceptance. No denial of that case would be possible
if the theory were indeed a law.

We conclude therefore — and the conclusion is vital
to the view of science presented here — that a theory is
not a law, and consequently, that the explanation afforded
by^tEeory cannot simply be the explanation by generaliza-
tion which consists in the exhibition of one law as a par-
ticular instance of another. It does not follow that
theories have nothing to do with laws, and that it is
immaterial for the theory that the laws of dynamics
are true, and of very great generality. We shall see
presently that this feature is of great importance. But
it does not involve that the theory is itself a law.

THE VALUE OF THEORIES

After this protest against a dangerous misunderstand-
ing, let us return and develop further our view of theories.

THE EXPLANATION OF LAWS 87

So far the truth of a theory has been based on two
grounds ; first, that the laws to be explained can be
deduced from it ; second, that it really explains in the
sense that has been indicated. But actually there is in
addition, a.tliirdjtest_pf the truth of a theory, which is
of great importance ; a true theory will not only explain
adequately the laws that it was introduced to explain ;
it will alsojDrfidict and «xplain~in-^vaiLce,laws.wiuch were
~unFnQwn before . All the chief theories in science (or
aTTeast in physics) have satisfied this test ; they have
all led directly to the discovery of new laws which were
unsuspected before the theory was proposed.

It is easy to see how a theory may predict new laws.
The theory, if it is worthy of consideration at all, will
be such that the old laws can be deduced from it. It may
easily be found on examination that not only these laws,
but others also can be deduced from it ; so far as the
theory is concerned, these others differ in no way from the
known laws, and if the theory is to be true, these laws
that are consequences of it must be true. As a matter of
fact, it is very seldom that a theory, exactly in its original
form, predicts any laws except those that it was proposed
to explain ; but a . very small and extremely natural
development of it may make it predict new laws. Thus,
to take our example, in order to explain the laws (Boyle's
and Gay-Lussac's) to which the theory was originally
applied, it is unnecessary to make any assumption about
the size of the molecules ; those laws can be deduced from
the theory whatever that size (so long as it is below a
certain limit) and the 'assumption was at first made for
simplicity that the molecules were mathematical points
without any size at all. But obviously it was more
natural to assume that the molecules, though extremely
small, l have some size and once that assumption is made,

1 If a drop of water were magnified to the size of the earth, the
molecules would be about the size of cricket balls.

88 WHAT IS SCIENCE?

laws are predicted which had not been discovered at
the time and would never have been suspected apart
from the theory. Thus, it is easy to see that, if the
molecules have a definite size, the behaviour of a gas,
when the number of molecules contained in a given vessel
is so great that the space actually occupied by the mole-
cules is nearly the whole of the space in the vessel, will
be very different from its behaviour when there are
so few molecules that practically all that space is unoccu-
pied. This expectation, a direct result of the theory, is
definitely confirmed by experiments which show a change
in the laws of a gas when it is highly compressed, and all
its molecules forced into a small volume.

This test of predicting new and true laws is always
applied to any theory when it is proposed. The first
thing we do when anyone proposes a theory to explain
laws, is to try to deduce from the theory, or from some
slight but very natural development of it, new laws,
which were not taken into consideration in the formulation
of the theory. If we can find such laws and prove by
experiment that they are true, then we feel much more
confidence in the theory ; if they are not true, we know
that the theory is not true ; but we may still believe that
a relatively slight modification will restore its value. It
is in this way that most new laws are actually suggested
for the purposes discussed in the previous chapter. At
the present time, in the more highly developed sciences,
it is very unusual for a new law to be discovered or sug-
gested simply by making experiments and observations
and examining the results (although cases of this character
occur from time to time) ; almost all advances in the
formulation of new laws follow on the invention of theories
to explain the old laws. Indeed it has been urged that
the only use of theories is thus to suggest laws among
which some will be found to be true. This opinion has
been much favoured by philosophers and mathematicians

THE EXPLANATION OF LAWS 89

and has always been accompanied by the opinion that it
is the end and object of science to discover laws. It
has also been professed (especially at the end of the
nineteenth century) by people who know something about
science and actually practised it ; but I think that these
people only professed the view because they were afraid
what the philosophers might say if they denied it. At
any rate, for myself, I cannot understand how anybody
can find any interest in science, who thinks that its task
is completed with the discovery of laws.

For the explanation of laws, though it is formally quite a
different process from the discovery of laws, is in its object
merely an extension of that process. We seek to discover I
laws in order to make nature intelligible to us ; we seek /
to explain them for exactly the same reason. The end
at which we are aiming in one process as in the other is
the reconciliation with out intellectual desires of the
perceptions forced on us by the external world of nature
What possible reason can be given for attaching immense
importance to one stage in the process and denying all
intrinsic importance to another ? Surely so long as
anything remains to be explained it is the business of
science to continue to seek explanations.

THE INVENTION OF THEORIES

And here again arises obviously a question very similar
to that discussed in the previous chapter. A theory, it
has been maintained, is some proposition which satisfies
these conditions : (i) It must be such that the laws whicf^
it is devised to explain can be deduced from it ; (2) it
must explain those laws in the sense of introducing ideas
which are more familiar or, in some other way, mord[
acceptable than those of the laws ; (3) it must predict!
new laws and these laws must turn out to be truer"T5f
course we have to ask now how such theories are w be

90 WHAT IS SCIENCE?

found ; we might find theories to satisfy the first two
conditions by the exercise of sufficient patience in a pro-
cess of trial and error ; but how can we possibly be sure
that they will satisfy the third ? The answer that
must be given will be clear to the reader if he has accepted
the results of the previous discussion. There cannot be,
from the very nature of the case, any kind of rule whereby
the third condition may certainly be satisfied ; the mean-
ing of the condition precludes any rule. [Sctualry theories
rafe~always suggested in view of the first two conditions
\only; and actually it turns out that they often fulfil
<;4he thirdj And once again they most often turn out to
fulfiUhe third when they are suggested by those excep-
tional individuals who are the great men of science. It
is when the theory seems to them to explain the laws,
when the ideas introduced by it appear to them acceptable
and satisfactory, that nature conforms to their desires,
and permits to be established by experiment the laws
which are the direct consequence of those ideas.

The form in which that statement is put may appear
rather extravagant, and we shall return later and consider
some questions which it seems to raise. But the general
view that true theories are the expression of individual
genius will probably seem less paradoxical than the view
put forward in the previous chapter, that true laws also
contain a personal element. The difference is indicated
by the words that are used ; we speak of the " discovery "
of a law, but of the " invention " of a theory. A law,
• it is implied, is something already existing which merely
| lies hidden until the discoverer discloses it ; a theory, on
I the other hand, does not exist apart from the inventor ;
it is brought into being by an effort of imaginative
thought, "jkj do not think that distinction will bear
examination ; it seems to me very difficult to regard laws
and not theories as something existing independently of
investigation and wholly imposed by the external world,

THE EXPLANATION OF LAWS 91

or theories and not laws as the product of the internal
world of the intellect. F"For both theories and laws derive
their ultimate value from their concordance with nature
and both arise from mental processes of the same kin3^
Moreover, as has been suggested already, in the more
highly developed sciences of to-day theories play a very
large part in determining laws ; they not only suggest
laws which are subsequently confirmed by experimental
investigation, but they also decide whether suggested
laws are or are not to be accepted. For, as our discussion
in the previous chapter showed, experiment alone cannot
decide with perfect definiteness whether or no a law is
to be accepted ; there are always loopholes left which
enable us to reject a law, however much experimental
evidence may suggest it and enable us to maintain a law
(slightly modified) even when experimental evidence seems
directly to contradict it. An examination of any actual
science will show that the acceptance of a law is very
largely determined by the possibility of explaining it
by means of a theory ; if it can be so explained, we are
much more ready to accept it and much more anxious
to maintain it than we should be if it were not the conse-
quence of some theory. Indeed many laws in science are
termed " empirical " and regarded with a certain amount
of suspicion ; if we inquire we find that an empirical law
is simply one of which no theoretical explanation is known.
In the science of physics at least, it would almost be more
accurate to say that we believe our laws because they are
consequences of our theories than to say that we believe
our theories because they predict and explain true laws !
On such grounds I reject the view (though it is generally
prevalent) .that laws are any less the product of imagina-
tive thought than ?rp fhrnrip.^ The, problem why nature
conforms with our intellectual desires arises just as clearly
with one as with the other. Nevertheless it is doubtless
true that the 'personal and imaginative element is more

92 WHAT IS SCIENCE?

obvious and more prominent in theories than in laws.
One aspect of this difference has been noted already ;
the acceptance of a theory as true does involve a personal
choice in a way that a law does not. Different people do
differ about theories ; they can choose whether or no they
will believe them ; but people do not differ about laws ;*
there is no personal choice ; universal agreement can be
forced. Again, if we look at the history of science, we shall
find that the great advances in theory are more closely
connected with the names of the great men than are the
advances in laws. Every important theory is associated
with some man whose scientific work was notable apart
from that theory ; either he invented other important
theories or in some other way he did scientific work
greatly above the average. On the other hand there are
a good many well-known laws which are associated with
the names of men who, apart from those particular
laws, are practically unknown ; they discovered one
important law, but they have no claim to rank among
the geniuses of science.2 That fact seems to indicate
that a greater degree of genius is needed to invent true
theories than to discover true laws.

The same feature appears in the early and prehistoric
stages of science. Science, as we have seen, originally
took over from common sense laws which had been
already elaborated ; and although it has greatly refined
and elaborated those laws and has added many new
types, it has never wholly abandoned the laws of common
sense. Modern science depends as much as the crudest
common sense on the notion of a " substance " (a notion

1 Except in so far as people may refuse to admit a law or to regard
it as anything but empirical, because it is not in accordance with
some theory. But then they admit that the laws describe the facts
rightly ; they only suggest that some other and equally accurate way
of describing them would be preferable.

a In case this book falls into the hands of some expert physicist,
I would suggest that examples may be found in the laws of Stefan,
Dulong and Petit, or Bode,

THE EXPLANATION OF LAWS 93

which, as we have seen, implies a law), on the notion of
the succession of events in time and the separation of
bodies in space and so on. But science has abandoned
almost all the theories of pre-scientific days. For there
were and are such non-scientific theories ; and it is
because the plain man has theories of his own, just as
much as the most advanced man of science, that it has
not been necessary to occupy a larger space in explaining
exactly what a theory is ; the reader will probably
have recognized at once something familiar in the kind
of explanation which the dynamical theory of gases
offers. The most typical theories of the pre-scientific
era were those which explained the processes occurring in
nature by the agencies of beings analogous to men — gods,
fairies, or demons. The " Natural Theology " of the
eighteenth century which tried to explain nature in
terms of the characteristics of a God, known through
His works, was a theory of that type ; in the features
which have been described as essential to theories it
differed in no way from that which we have discussed.
But all such theories have been abandoned by science ;
the theories that it employs are of a type quite unknown
before the seventeenth century.1 In respect of theories
science has diverged completely from common sense ;
and the divergence can be traced accurately to the
work of a few great men. Common sense is therefore
more ready to accept theories rather than laws as the
work of individual genius.

But while I accept fully the view that the formulation
of a new theory, and especially of a new type of theory,

1 An exception is often made in favour of Lucretius, who wrote
about 70 B.C. But my own opinion is that moderns, with their fuller
knowledge, read into his works ideas which never entered the head
of their author. I do not think that (as Mr. Wells maintains in his
" Outline of History ") it was merely the barrenness of the soil on which
his seed fell that prevented it blossoming into fruit. The sterility of
his ideas, contrasted with those of Galileo and Newton, was inherent
in them.

94 WHAT IS SCIENCE?

is a greater achievement than the formulation of a new
law, I cannot admit that the two processes are essentially
different. As Galileo was the founder of experimental
physics, so Newton was the founder of theoretical physics ;
as Galileo first introduced the type of law which has
become most characteristic of the science, so Newton
introduced first the characteristic type of theory. And
of the two Newton is rightly judged by popular opinion
to have been the greater man. But it is not rightly
recognized how great was the acnievement of Galileo ;
indeed his fame is usually associated with things — the
observation of the isochronism of the pendulum, or his
fight with clericalism over the Copernican theory —
other than with his greatest service to science. It is
his establishment of the first experimental numerical
law that constitutes his highest claim to greatness, and
that law was as much an expression of his personality
as the theory of Newton.

THE ANALOGIES OF THEORIES

Mention has just been made of " types " of theories.
There are such types, just as there are types of laws,
and they play the same role in permitting lesser men to
complete and extend the work of the greater. Once a
theory of a new type has been invented and has been
shown to be true in the explanation of laws, it is naturally
suggested that similar theories may prove equally
successful in the explanation of other laws. And on the
whole the suggestion has proved true. In each branch
of science there are certain very broad and general
theories which have been invented by the founders of that
branch ; subsequent development of that branch usually
consists largely in the ampliation and slight modifica-
tion of such fundamental theories by investigators,
many of whom could never have themselves laid the

THE EXPLANATION OF LAWS 95

foundation. Indeed, the investigator often feels that in
finding an explanation for the laws that he has discovered
he has little more latitude than in discovering those
laws ; it is perfectly clear from the outside what kind of
theory he must seek, just as it is clear what kind of law
he must seek.

Thus, it may be stated broadly1 that from 1700 to 1870
all physical theories were of a single type of which the
dynamical theory of gases which we have used as an
example provides an excellent instance. They were
all " mechanical ._" theories. In our example, the
behaviour of gases is explained by an analogy with a
piece of mechanism, a set of moving parts reacting
on each other with forces which determine and are
determined by the motion. That feature is common to
all the mechanical theories which played so great a
part in the older physics and are still prominent in the
newer ; ^tSey explain laws by tracing an analogy between
the system of which the laws are to be explained and
some piece of mechanisnV] Once it was realized that
such theories were likely to turn out to be true, the
task of inventing theories was greatly simplified ; it
often became simply that of devising a piece of mechanism
which would simulate the behaviour of the system of
which the laws were to be explained.

But all scientific theories are not mechanical. In
physics it is the admission of theories that do not fall
within this class which distinguishes the newer from the
older study. And in other branches of science (except
where they are obviously founded on physics) theories
of other kinds are the rule. For instance, the theory
of evolution, proposed to explain the diversity and yet
the resemblances of different kinds of living beings is

1 A very important exception must be made of the purely mathe-
matical theories, such as those of Fourier and Ampere. Some indica-
tion of the nature of these theories is given in Chapter VII.

96 WHAT IS SCIENCE?

not mechanical ; it does not trace an analogy between the
production of such beings and the operation of a piece
of mechanism. Can we find any feature that is common
to all theories that have proved to be true, or must we
(as in the case of laws) rest content with several distinct
but well-defined types which have all proved successful
and yet display no common characteristic ?

I think we can find such a feature. The explanation
offered by a theory (that is to say, the part of the theory
which does not depend simply on the deduction from it
of the laws to be explained) is always based on an analogy,
and the system with which an analogy is traced is always
one of which the laws are known ; it is always one of
those systems which form part of that external world
of which the subject-matter of science consists. The
theory always explains laws by showing that if we
imagine that the system to which those laws apply
consists in some way of other systems to which some
other known laws apply, then the laws can be deduced
from the theory. Thus our theory of gases explains
the laws of gases on the analogy of a system subject
to dynamical laws. The theory of evolution explains
the laws involved in the assertion that there are such-
and-such living beings by supposing that these living
beings are the descendants of others whose characters
have been modified by reaction to their surroundings
in a manner which is described by laws applicable to
living beings at the present day. Again the immense
theory involved in the whole science of geology explains
the structure of the earth as it exists to-day by supposing
that this structure is the result of the age-long operation
of influences, the action of which is described by laws
observable in modern conditions. Q In each case the
" explaining " system is supposed to operate according
to known laws, but it is not a system of which those
laws can be asserted as laws, because it is, by the very

THE EXPLANATION OF LAWS 97

supposition underlying the theory, one which could never
be observed — either because it is too small or too remote
in the past or for some similar reason — and therefore
does not form part of the proper subject-matter of science.
It is because the explanation offered by a theory is
always based on an analogy with laws that the distinction
between laws and theories has been so often overlooked.
The statement of the dynamical theory of gases about
the properties and behaviour of molecules, or of the theory
of evolution about the process whereby the existing
species of living beings came into existence, is so similar
to the statement, asserted by a law, about the properties
of actual mechanical 'systems or about the changes that
are proceeding in existing species that the vital difference
between the two is forgotten. The statement asserted
by a law can be proved by direct perception ; it states
something which can be observed and which can be the
subject of universal assent. The statement involved
in the theory cannot be proved by direct perception, for
it does not state anything that can be or has been observed.
The failure to observe this distinction and the conse-
quent failure to give to theories their true place in the
scheme of science is the cause of most of the misunder-
standings that are so widely prevalent concerning the
nature and objects of science. For it has been admitted
that, though the discovery of laws depends ultimately
not on fixed rules but on the imagination of highly
gifted individuals, this imaginative and personal element
is much more prominent in the development of theories ;
the neglect of theories leads directly to the neglect of
the imaginative and personal element in science. It
leads to an utterly false contrast between " materialistic "
science and the " humanistic " studies of literature,
history and art.

98 WHAT IS SCIENCE?

SCIENCE AND IMAGINATION

At the risk of wearying the reader by endless repetition
I have insisted on the fallacy of neglecting the imaginative
element which inspires science just as much as art. If
this book is to fulfil any useful purpose that insistence is
necessary. For it is my object to attract students to
science and to help them to understand it by showing
them from the outset what they may expect from it.
It is certain that one of the chief reasons why science
has not been a popular subject in adult education, and is
scarcely recognized even yet as a necessary element of
any complete education, is the impression that science
is in some way less human than other studies. And for
that impression men of science are themselves more to
blame than anyone else ; in a mistaken endeavour to
exalt the certainty of their knowledge they deliberately
conceal that, like all possible knowledge, it is personal.
They exhibit to the outside world only the dry bones of
science from which the spirit has departed.

It is true that it is less easy for the beginner to grasp
the imaginative element in science than in some other
studies. A larger basis of mere information is perhaps
required before it becomes apparent. And, of course,
he can never hope to share himself the joy of discovery ;
but in that respect he is no worse off than many who make
of science their life-work. But he can, if he will take the
trouble, appreciate the discoveries of others and experi-
ence at second-hand the thrill of artistic creation. For
those who have the necessary knowledge, it is as exciting
to trace the development of a great scientific theory,
which we could never have developed ourselves, as it is
to -read great poetry or to hear great music which we
ourselves could never have written. But I must admit
that the books on science are few which make it easy
for the beginner to share that experience. And so,

THE EXPLANATION OF LAWS 99

although I can hardly hope that I shall succeed where
so many writers have failed, and although the attempt
transgresses the strict limit of an introduction, I should
like to try to tell again the familiar story of one of the
most wonderful romances of science — the story of
Newton and the apple I1

The early chapters of the story must be greatly abbrevi-
ated. Copernicus and Kepler, a century before Newton,
had shown clearly what were the paths in which the
planets move about the sun and the satellites, such as
our moon, about the planets. It does not seem clear
whether anyone before Newton had thought of inquiring
why they should move in such paths, or had ever con-
templated the possibility of explaining the laws which
Kepler had laid down. In science, as in many other
things, it is often much harder to ask questions than to
answer them. People might have said, and many
probably did say : The planets have to move somehow ;
the paths Kepler describes are quite simple ; why should
not the planets move in them ? It is as ridiculous to ask
why they so move as to ask why a man's hair is yellow
or brown or red, and not blue or green. The mere con-
ception of explaining the paths of the planets was itself
an immense achievement.

And we can see now what suggested it to Newton.
Some sixty years before Galileo had, for the first time,
discovered some of the laws which govern the motions of
bodies under forces. He had shown that, in some simple
instances at least, there were such things as " laws of
dynamics." The idea occurred to Newton, May not the
movements of the planets and their satellites be subject
to just such laws of dynamics as Galileo had discovered
for the ordinary bodies which we see and handle. If so,

1 Of course the apple may be mythical — like all the historical
objects of our childhood. And it is impossible to be certain what
Newton really thought. But his thought might have followed the
line suggested here.

100 WHAT IS SCIENCE?

we ought to be able to find a set of forces such that if they
act on the planets according to Galileo's laws, the planets
will move as Kepler has shown that they do move. That
seems to us very obvious now ; but it was not obvious
then. Galileo, as far as we know, never thought of it ;
nor did anyone in the two generations between him 'and
Newton. And perhaps one reason why nobody thought
of it was that they realized instinctively that if they had
thought of it they could have got no further. To-day any
clever schoolboy could solve the next problem which
presents itself, namely that of finding what forces, acting
according to Galileo's laws, would make the planets move
as they do ; but that is only because Newton has shown
him the way. In order to solve this problem which seems
to us now so easy, Newton had to invent modern mathe-
matics ; he had to make a greater advance in mathe-
matics than had been made in all the time since the high-
water of Egyptian civilization. This achievement of
his was quite as wonderful as any other ; but as it was
not characteristically scientific (in the modern sense) it
may be left on one side here.

So he solved his problem. He showed that the planets
can be regarded as subject to Galileo's laws, and that the
force on a planet must be directed towards the sun, and
that on the moon towards the earth ; and that these forces
must vary in a certain simple way with the distance
between planet and sun or moon and earth. The moon
follows the course she does because there is a pull between
it and the earth just as, when a stone is whirled round at
the end of a string, there is a pull between the stone and
the hand.

And now — as I like to think — he had ended his labour.
He realized that he had made a stupendous discovery,
which must revolutionize, as it actually has done, the
whole science of astronomy. He had shown that the
laws of dynamics apply to planets as well as to ordinary

THE EXPLANATION OF I'AWS

bodies on the earth, tnat the planets were subject to
forces, and had determined what those forces were.
What more could he do ? What explanation of his result
could be offered or even demanded. He had ordered the
solar system, and who could be so foolish as to ask why
the order was that which he had found and no other ?
But after his morning's work which finally completed his
forthcoming treatise on the matter, he sat in his orchard
and was visited by some of his friends from Cambridge.
Perhaps they too were natural philosophers and he talked
to them about what he had been doing ; but it is more
likely that they sat idle, talking the thin-blooded intel-
lectual scandal that must have always flourished in
academic society, while Newton played with the historic
kitten.

And then the apple fell from the tree. Newton was
suddenly silent in a reverie ; the kitten played unheeded
with the fallen fruit ; and his friends, used to such sudden
moods, laughed and chattered. After a few moments he
must have paper ; he rushes to his desk in the house ;
scribbles a few hasty figures ; and now the theory of
gravitation is part of the structure of the universe. The
falling apple, a trivial incident which he had seen a
thousand times before, had loosed a spring in his mind,
set unconsciously by all his previous thought. He had
never consciously asked himself why the moon was pulled
towards the earth until, in an instant of revelation, the
apple appeared to him not " falling " (the meaningless
word he had always used before), but pulled towards the
earth. The idea flashed on him quicker than it could
be spoken. If both the moon and the apple are pulled
towards the earth, may they not be pulled by the same
force ? May not the force that makes the apple " fall "
be that which restrains the moon in its orbit ? A simple
calculation will test the idea. He knows how far the
moon is from the earth and how the force on it varies

102 WHAT IS i CIENCE ?

with the distance and with the size of the moon. If
the moon were brought to the surface of the earth and
reduced to the size of the apple would the force on it be
such that it would fall with the speed of the apple ? The
answer is, Yes ! x The motion of the planets is therefore
explained both by generalization and by familiarity.
That motion is merely one particular instance of a general
principle of which the very familiar fall of heavy objects
is another.

What I want to impress on the reader is how purely
personal was Newton's idea. His theory of universal
gravitation, suggested to him by the trivial fall of an
apple, was a product of his individual mind, just as much
as the Fifth Symphony (said to have been suggested by
another trivial incident, the knocking at a door) was a
product of Beethoven's. The analogy seems to me exact.
Beethoven's music did not exist before Beethoven wrote
it, and Newton's theory did not exist before he thought
of it. Neither resulted from a mere discovery of some-
thing that was already there ; both were brought into
being by the imaginative creation of a great artist. HOWT-
ever there is one apparent difference ; Beethoven having

1 The reader who knows the story — and who does not ? — will see that
here I deviate widely from history. Newton did not know how far
the moon was from the earth ; current estimates were wrong ; and at
first he was therefore doubtful of his theory. But when the distance
was measured more accurately, he found it agreed perfectly with his
theory. I hesitate to suggest that it was Newton's theory that had
changed the distance !

I feel, too, that some people will think that I must be very antiquated
in my knowledge if I glorify Newton so greatly when the daily Press
has been assuring us lately that his ideas have been completely over-
thrown by Einstein. This is not the place to discuss what Einstein
has proved. 1 admire his work as much as anyone, but he has not
invalidated in the smallest degree the great discovery of Xewton
which is discussed in the text. It is still as certain as ever it was that
the fall of the apple and the " fall " of the moon are merely two
examples of the very same fundamental principle ; and it is as certain
as ever that the motions of the planets are subject to the same laws
as those of terrestrial bodies. What is now not quite certain is whether
Galileo's laws are strictly applicable to circumstances very different
to those of the experiments by which he proved them,

THE EXPLANATION OF LAWS 103

conceived his symphony had no need to test it in order
to discover if it was " right," while Newton had to com-
pare the results of his theory with the external world,
before he was sure of its true value. Does not this show
that Newton's achievement was not so perfectly personal
and imaginative as Beethoven's ?

I do not think so. First, Beethoven's work had to be
tested ; the test of artistic greatness is appeal to succeed-
ing generations free from the circumstances in which the
work was conceived ; it is very nearly the test of universal
agreement. But it is another point of view I want to
emphasize here. It is said that Newton's theory was
not known to be true before it was tested ; but Newton
knew that it was true — of that I am certain. To our
lesser minds there seems no imperative reason why the
force on the moon and the force on the apple should be
related as closely as the theory of gravitation demands,
merely because it would be so delightfully simple if they
were ; but Newton probably felt no doubt whatever on
the matter. As soon as it had occurred to him that the
fall of the apple and the fall of the moon might be the
same thing, he was utterly sure that they were the same
thing ; so beautiful an idea must be true. To him the
confirmation of numerical agreement added nothing to
the certainty ; he examined whether the facts agreed
with the object of convincing others, not himself. And
when the facts as he knew them did not agree, we may be
sure that his faith in the theory was in no way shaken ;
he knew that the facts must be wrong, but he had to wait
many years before evidence of their falsity was found
which would appeal to those who had not his genius and
could not be sure of harmony between their desires and
the external world.

^ WHAT IS SCIENCE?

ARE THEORIES REAL ?

And here we come to our last question. I have been
at pains to distinguish theories from laws, and to insist
that theories are 'not laws. But if that contention is
true, are not theories deprived of much of their value ?
Laws, it may be said, are statements about real things,
about real substances (such as iron), about real objects
(such as the earth or the planets or existing living beings).
Laws are valuable because they tell us the properties of
these real objects. But if theories are not laws, and if the
statements they make are about things that cannot ever
be the subject of laws, do they tell us about anything
real ? Are the molecules (by means of which we explain
the properties of gases) or the countless generations of
unknown animals and plants (by means of which we
explain the connexions between known animals and
plants) or the forces on the planets (by means of which
we explain their orbit) — are these molecules and animals
and forces mere products of our fantasy, or are they just
as real as the gases and the animals the laws of which
they are led to explain ? Are theories merely explana-
tory, are they like the fairy tales by means of which our
ancestors explained to themselves the world about them,
are they like the tales we often tell to our children with
the same object of explanation, or are they truly solid
fact about the real things of the world ?

That may seem a simple question to which a plain
answer, Yes or No, might be given; but in truth it
raises the most profound and abstruse problems of
philosophy and really lies without the scope of this book.
Our object is to discover what science is ; we have learnt
what laws and theories are, and what part they play in
science ; it is not directly part of our purpose to discuss
what value all this elaboration has when it is achieved.
But in a book of this kind it would be wrong to leave the

THE EXPLANATION OF LAWS 105

question with no answer ; and I will therefore explain
how the matter appears to me, although I know that
many other people would give different answers.

I should reply to the questioner by asking him what he
means by " real " and why he is so sure that a piece of
iron, or a dog, is a real object. And the answer that I
should suggest to him is that he calls these things real
because they are necessary to make the world intelligbile
to him ; and that it is because they are necessary to make
the world intelligible to him that he resents so strongly
(as he will if he is a plain man) the suggestions that some
philosophers have made that these things are not real.
It is true that these suggestions are often not interpreted
rightly, and that what the philosophers propose is not
so absurd as appears at first sight ; but the fact remains
that these ideas are of supreme importance to him in
making the world intelligible, and that he dislikes the
notion that they are in any sense less valuable than other
ideas which, for him at least, do not make the world so
intelligible. The invariable associations which are
implied by the use of the ideas " iron " and " dog " are
extremely important in all his practical life ; it is
extremely important for him that a certain hardness and
strength and density and so on are invariably associated
in the manner which we assert when we say that there is
iron, and that a certain form and sound and behaviour
are invariably associated in the way that we assert when
we say that there are dogs. When the plain man says
iron and dogs are real objects he means (I suggest) to
assert that there are such ' invariable associations, that
they are extremely important, and that they are rendered
intelligible only by the assertion that there is iron and
there are dogs.

If we accept this view it is clear that we must answer

in_tjie affirmative the question from which we started.

fTheories are also designed to make the world intelligible

106_ WHAT IS SCIENCE?

to us, and they play quite as important a part as do laws
in rendering it intelligible. And if anything is real that
renders the world intelligible, then surely the ideas of
theories — molecules and extinct animals and all the rest
of it — have just as much claim to reality as the ideas of
laws.

But my questioner will almost certainly not be satisfied
with that answer ; it will seem to him to shirk just the
question that he wants to raise. He will feel that the
view that reality is merely what leads to intelligibility
deprives reality of all its importance ; if science is merely
an attempt to render the world intelligible, in what does
it differ from a fairy tale — which has often the same
object ? Or — to put the matter in a different way —
intelligibility is a quality that depends on the person who
understands ; one person may find intelligible what
another may not. Reality on the other hand is, by its
very meaning, something independent of the person who
thinks about it. When we say that a thing is real we do
not mean that it is peculiarly suited to our understand-
ing ; we mean much more that it is something utterly
independent of all understanding ; something that would
be the s?me if nobody ever thought about it at all or ever
wanted to understand it.

I think the essence of this objection lies in the sentence :
" One person may find intelligible what another may not."
When we feel that science is deprived of all value by being
likened to a fairy tale, our reason is that different people
like different fairy tales, and that one fairy tale is as good
as another. But what if there were only one possible
fairy tale, only one which would explain the world, and
if that one were intelligible and satisfactory to every
one ? For that is the position of science. There have
been many fairy tales to explain the world ; every myth
and every religion is (in part at least) a fairy tale with that
object. But the fairy tale which we call science differs

THE EXPLANATION OF LAWS 107

from these in one all-important feature ; it is the fairy
tale which appeals to every one, and the fairy tale which
nature has agreed to accept. It is not you and I and the
man round the corner who find that the conception of
iron makes the world intelligible, while the people in the
next street do not ; in this matter every single living being
in the world (so far as we can ascertain his opinion)
agrees with us ; they all accept our fairy tale and agree
that it makes the world intelligible. And nature accepts
it too ; the law that there is iron enables us to predict,
and nature always agrees with our predictions. There
is no other fairy tale like this ; there is none that denies
that there is iron, a substance with invariably associated
properties, which is acceptable to every one and which
predicts truly. It is just because our fairy tale is capable of
the universal agreement which we discussed in Chapter II
that we distinguish it from all other fairy tales and
call it solid fact. Nevertheless the fact remains that its
value for us is that of other fairy tales, namely that it
makes the world intelligible.

And now let us turn again to theories. Here, it is true,
we cannot apply directly the criterion of universal assent.
There is actually much more difference of opinion concern-
ing the value of theories than there is concerning the value
of laws ; and it is impossible to force an agreement as it
can be forced in the case of laws. And while that
difference of opinion persists we must freely admit that
the theory has no more claim on our attention than any
other ; it is a fairy tale which may be true, but which is
not known to be true. But in process of time the
difference of opinion is always resolved ; it vanishes
ultimately because one of the alternative theories is
found to predict true laws and the others are not. It is
for this reason that prediction by theories is so
fundamentally important ; it enables us to distinguish
between theories and to separate from among our fairy

108 WHAT IS SCIENCE?

tales that one which nature is prepared to accept and can
therefore be transferred from the realm of fantasy to
that of solid fact. And when a theory has been so trans-
ferred, when it has gained universal acceptance because,
alone of all possible alternatives, it will predict true laws,
then, although it has purpose and value for us because
it renders the world intelligible, it is so clearly distin-
guished from all other attempts to achieve the same pur-
pose and to attain the same value that the ideas involved
in it, like the ideas involved in laws, have the certainty
and the universality that is characteristic of real objects.
A molecule is as real, and real in the same way, as the gases
the laws of which it explains. It is an idea essential to
the intelligibility of the world not to one mind, but to
all ; it is an idea which nature as well as mankind accepts.
That, I maintain, is the test and the very meaning of
reality.

CHAPTER VI
MEASUREMENT

WE have now examined the chief types of scientific
proposition and discussed what are the prin-
ciples and the facts on which all science rests.
Already we have had occasion to notice differences
between the various branches of science, and when we
leave such very fundamental questions the differences
are bound to become more prominent. It does not seem
to me that there is much left to say (except in amplifica-
tion of what has been said already) that would be applic-
able to the whole of science. But there is one further
matter which may fitly receive some attention ; for
though it affects only part of science, that part is con-
stantly growing both in volume and in importance.
Moreover, the sciences into which it enters are generally
held to be particularly difficult to popularize, and to be
beyond the reach of the untrained reader. Accordingly,
we shall scarcely diverge from the main purpose of our
discussion if in the next two chapters we give it some
consideration.

This matter is measurement and all the structure of
mathematical science which rests upon measurement.
Every one knows that measurement is a very important
part of many sciences ; they know too, that many sciences
are " mathematical " and can only be apprehended
completely by those versed in mathematics. But very
few people could explain exactly how measurement enters
into science, why it enters into some and not in others,
why it is so important, what mathematics is and why it
is so intimately connected with measurement and with
the sciences in which measurement is involved. In the

109

110 WHAT IS SCIENCE ?

next two chapters I propose to attempt some answer to
these questions. Any answer to these questions that
can be attempted here will not, of course, enable any-
one to start immediately the study of one of the mathe-
matical sciences in the hope of understanding it
completely. But if he can be convinced that even in
the most abstruse parts of those sciences there is
something that he can comprehend and appreciate
without the smallest knowledge of mathematics, some-
thing may be done towards extending the range of
the sciences that are open to the layman.

WHAT IS MEASUREMENT ?

Measurement is one of the notions which modern
science has taken over from common sense. Measure-
ment does not appear as part of common sense until
a comparatively high stage of civilization is reached ;
and even the common-sense conception has changed
and developed enormously in historic times. When I
say that measurement belongs to common sense, I only
mean that it is something with which every civilized
person to-day is entirely familiar. It ma^LJe defined,
in. general, as the assignment of numbers Jto_jrepresent
properties? If we say that the time is 3 o'clock, that
the price of coal is 56 shillings a ton, and that we have
just bought 2 tons of it — in all such cases we are using
numbers to convey important information about the
" properties " of the day, of coal in general, of the coal
in our cellar, or so on ; and our statement depends
somehow upon measurement.

The first point T want to notice i& that it is-only, some
properties,. and not all that can be thus represented by
numbers. If I am buying a sack of potatoes I may ask
what it weighs and what it costs ; to those questions
I shall expect a number in answer ; it weighs 56 Ibs.

MEASUREMENT 111

and costs 5 shillings. But I may also ask of what variety
the potatoes are, and whether they are good cookers ;
to those questions I shall not expect a number in answer.
The dealer may possibly call the variety "n.No. 11 " in
somebody's catalogue ; but even if he does, T shall feel
that such use of a number is not real measurement, and
is not of the same kind as the use in connexion with
weight or cost. What is the difference ? Why are
some properties measurable and others not ? Those
are the questions I want to discuss. And I will outline
the answer immediately in order that the reader may
see at what the subsequent discussion is aiming. The
difference is this. LSuppose I have two sacks of potatoes
which are identical in weight, cost, variety, and cooking
qualities ; and that I pour the two sacks into one so that
there is now only one sack of potatoes. This sack will
differ from the two original sacks in weight and cost
(the measurable properties), but will not differ from them
in variety and cooking qualities (the properties that
are not measurable). £TRe measurable properties of a
body are those which are changed by the combination
of similar bodies ; the non-measurable properties are
those that are not changed?? We shall see that this
definition is rather too crude, but it will serve for the
present.

NUMBERS

In order to see why this difference is so important
we must inquire more closely into the meaning of
" number." And at the outset we must note that
confusion is apt to arise because that word is used to
denote two perfectly different things. It sometimes
means a mere name or word or symbol, and it sometimes
means a property of an object. Thus, besides the
properties which have been mentioned, the sack of

112 WHAT IS SCIENCE?

potatoes has another definite property, namely the
number of potatoes in it, and the number is as much a
property of the object which we call a sack of potatoes
as its weight or its cost. This property can be (and
must be) " represented by a number " just as the weight
can be ; for instance, it might be represented by 200.
But this " 200 " is not itself a property of the sack ;
it is a mere mark on the paper for which would be sub-
stituted, if I was speaking instead of writing, a spoken
sound ; it is a name or symbol for the property. .When
we say that measurement is the representation of
properties by " imbibers/' we mean that it Is the repre-
sentation of properties, other than number, by the
are alway^ used to represent number.

Moreover, there is a separate word for these symbols ;
they are called " numerals." We shall always use that
word in future and confine " number " to the meaning
cf the property which is always represented by numerals.
These r.rmsi derations are not mere quibbling over
V>Hng n^ _ dearly an irqpnrta.nt point.

_

ijarnely, tfra,t the measurable properties of an pbj ect
must resemble in some special way the property number,
since they can be fitly represented by the same symbols ;
they must have some quality common with number.
We must proceed to ask what this common quality is,
and the best way to begin is to examine the property
number rather more closely.

The number of a sack of potatoes, or, as it is more
usually expressed, the number of potatoes contained
in it, is ascertained by the process of counting. Count-
ing is inseparably connected in our minds to-day with
numerals, but the process can be, and at an earlier stage
of civilization was, carried on without them. Without
any use of numerals I can determine whether the number
of one sack of potatoes is equal to that of another. For
this purpose I take a potato from one sack, mark it in

MEASUREMENT 113

some way to distinguish it from the rest (e.g. by putting
it into a box), and then perform a similar operation on
a potato from the other sack. I then repeat this double
operation continually until I have exhausted the potatoes
from one sack. If the operation which exhausts the
potatoes from one sack exhausts also the potatoes from
the other, then I know that the sacks had the same
number of potatoes ; if not, then the sack which is not
exhausted had a larger number of potatoes than the
other.

THE RULES FOR COUNTING

This process could be applied equally well if the objects
counted against each other were not of the same nature.
The potatoes in a sack can be counted, not only against
another collection of potatoes, but also against the men
in a regiment or against the days in the year. The
" mark/' which is used for distinguishing the objects
in the process of counting, may have to be altered to suit
the objects counted, but some other suitable mark could
be found which would enable the process to be carried
out. If, having never heard of counting before, we applied
the process to all kinds of different objects, we should
soon discover certain rules which would enable us to
abbreviate and simplify the process considerably. These
rules appear to us to-day so obvious as to be hardly
worth stating, but as they are undoubtedly employed
in modern methods of counting, we must notice them
here. The first is that if two sets of objects, when counted
against a third set, are found *to have the same number
as that third set, then, when counted against each other
they will be found to have the same number. This rule
enables us to determine whether two sets of objects
have the same number without bringing them together ;
if I want to find out whether the number of potatoes

8

114 WHAT IS SCIENCE?

in the sack I propose to buy is the same as that in a
sack I have at home, I need not bring my sack to the
shop ; I can count the potatoes at the shop against
some third collection, take this collection home, and
count it against my potatoes. Accordingly the discovery
of this first rule immediately suggests the use of portable
collections which can be counted, first against one collec-
tion and then against another, in order to ascertain
whether these two have the same number.
-TT The value of this suggestion is increased greatly by
|J the discovery of a second rule. It is that by starting with
a single object and continually adding to it another single
object, we can build up a series of collections of which
one will have the same number as any other collection
whatsoever. This rule helps us in two ways. First,
since it states that it is possible to make a standard
series of collections one of which will have the same
number as any other collection, it suggests that it might
be well to count collections, not against each other, but
against a standard series of collections. If we could
carry this standard series about with us, we could always
ascertain whether any one collection had the same number
as any other by observing whether the member of the
standard series which had the same number as the first
had also the same number as the second. Next, it shows
us how to make such a standard series with the least
possible cumbrousness. If we had to have a totally
different collection for each member of *the standard
series, the whole series would be impossibly cumbrous ;
but our rule shows that the earlier members of the series
(that is those with the smaller number) may be all
parts of the later members. Suppose we have a collec-
tion of objects, each distinguishable from each other,
and agree to take one of these objects as the first member
of the series ; this object together with some other as
the next member ; these objects with yet another as

MEASUREMENT 115

the next member ; and so on. Then we shall obtain,
according to our rule, a series, some member of which
has the same number as any other collection we want to
count, and yet the number of objects, in all the members
of the standard series taken together, will not be greater
than that of the largest collection we want to count.

And, of course, this is the process actually adopted.
For the successive members of the standard series com-
pounded in this way, primitive man chose, as portable,
distinguishable objects, his fingers and toes. Civilized
man invented numerals for the same purpose. Numerals
are simply distinguishable objects out of which we build
our standard series of collections by adding them in turn
to previous members of the series. The first member of
our standard series is i, the next i, 2, the next i, 2, 3
and so on. We count other collections against these
members of the standard series and so ascertain whether
or no two collections so counted have the same number.
By an ingenious convention we describe which member
of the series has the same number as a collection counted
against it by quoting simply the last numeral in that
member ; we describe the fact that the collection of the
days of the week has the same number as the collection
i, 2, 3, 4, 5, 6, 7, by saying " thav. the number " of the
days of the week is 7. But when we say that what we
really mean, and what is really important, is that this
collection has the same number as the collection of
numerals (taken in the standard order) which ends in
7 and the same number as any other collection which
also has the same number as the collection of numerals
which ends in 7. 1

1 Numerals have also an immense advantage over fingers and toes
as objects of which the standard series may be formed, in that the
series can be extended indefinitely by a simple rule which automatically
gives names to any new numerals that may be required. Even if we have
never hitherto had reason to carry the series beyond (say) 131679
in order to count all the collections we have met with, when we do
meet at last with a larger collection, we know at once that the objects

116 WHAT IS SCIENCE?

The two rules that have been mentioned are necessary
to explain what we mean by " the number " of a collec-
tion and how we ascertain that number. There is a
third rule which is of great importance in the use of
numbers. We often want to know what is the number
of a collection which is formed by combining two other
collections of which the numbers are known, or, as it is
usually called, adding the two collections. For instance
we may ask what is the number of the collection made
by adding a collection of 2 objects to a collection of 3
objects. We all know the answer, 5. It can be found by
arguing thus : The first collection can be counted against
the numerals i, 2 ; the second against the numerals
i, 2, 3. But the numerals i, 2, 3, i, 2, a collection formed
by adding the two first collections, can be counted against
I> 2> 3» 4> 5- Therefore the number of the combined
collection is 5. However, a little examination will show
that in reaching this conclusion we have made use of
another rule, namely that if two collections A and a,
have the same number, and two other collections
B and b, have the same number, then the collection
formed by adding A to B has the same number as that
formed by adding a to b ; in other words, equals added to
equals produce equal sums. This is a third rule about
numbers and counting ; it is quite as important as the
other two rules ; all three are so obvious to us to-day
that we nevei think about them, but they must have
been definitely discovered at some time in the history of
mankind, and without them all, our habitual use of
numbers would be impossible.

we must add to our standard series arc 131680, 131681, and so on.
This is a triumph of conventional nomenclature, much more satis-
factory than the old convention that when we have exhausted our
fingers we must begin on our toes, but it is not essentially different.

MEASUREMENT 117

WHAT PROPERTIES ARE MEASURABLE ?

And now, after this discussion of number, we can return
to the other measurable properties of objects which, like
number, can be represented by numerals.} We can now
say more definitely what is the characteristic of these
properties which makes them measurable. It is that
there are rules true of these properties, closely analogous
to the rules on which the use of number depends. If a
property is to be measurable it must be such that (i)
two objects whicLare the same in respect of that property
as some third object are the same as each other ; (2) by
adding objects successively we must be able to make a
standard series one member of which will be the same in
respect of the property as any other object we want to
measure ; (3) equals added to equals produce equal sums.
In order to make a property measurable we must find
some method of judging equality and of adding objects,
such that these rules are true.

Let me explain what is meant by using as an example
the measurable property, weight.

Weight is measured by the balance. Two bodies are
judged to have the same weight if, when they are placed
in opposite pans, neither tends to sink ; and two bodies
are added in respect of weight when they are both placed
on the same pan of the balance. With these definitions
of equality and addition, it is found that the three rules
are obeyed, (i) If the body A balances the body B, and
B balances C, then A balances C. (2) By placing a body
in one pan and continually adding it to others, collections
can be built up which will balance any other body placed
in the other pan. 1 (3) If the body A balances the body B,
and C balances D, then A and C in the same pan will
balance B and D in the other pan. To make the matter
yet clearer let us take another measurable property,

1 See further, p. 122,

118 WHAT IS SCIENCE?

length. Two straight rods are judged equal in length, if
they can be placed so that both ends of one are contiguous
to both ends of the other ; they are added in respect of
length, when they are placed with one end of one con-
tiguous with one end of the other, while the two form
a single straight rod. Here again we find the three rules
fulfilled. Bodies which are equal in length to the same
body are equal in length to each other. By adding
successively rods to each other, a rod can be built up
which is equal to any other rod. And equal rods added to
equal rods produce equal rods. Length is therefore a
measurable property.

It is because these rules are true that measurement of
these properties is useful and possible ; it is these rules
that make the measurable properties so similar to
numbers, that it is possible and useful to represent them
by numerals the primary purpose of which is to repre-
sent numbers. It is because of them that it is possible
to find one, and only one numeral, which will fitly repre-
sent each property ; and it is because of them, that these
numerals, when they are found, tell us something useful
about the properties. One such use arises in the combina-
tion of bodies possessing the properties. We may want
to know how the property varies when bodies possessing
it are added in the way characteristic of measurement.
When we have assigned numerals to represent the
property we shall know that the body with the property
2 added to that with the property 3 will have the same
property as that with the property 5, or as the combina-
tion of the bodies with properties 4 and i. This is not
the place to examine exactly how these conclusions are
shown to be universally valid ; but they are valid only
because the three rules are true.

MEASUREMENT 119

THE LAWS OF MEASUREMENT

But what is the nature of these rules ? They are laws
established by definite experiment. The word " rule"
has been used hitherto, because it is not quite certain
whether they are truly laws in their application to
number ; but they certainly are laws in their application
to other measurable properties, such as weight or length.
Xhe fact that the rules-are true can be, and must be, deter-
mined by experiment in the same way as the fact that
any other laws are true. Perhaps it may have appeared
to the reader that the rules must be true ; that it requires
no experiment to determine that bodies which balance
the same body will balance each other ; and that it
is inconceivable that this rule should not be true. But I
think he will change his opinion, if it is pointed out that
the rule is actually true only in certain conditions ; for
instance, it is only true if the balance is a good one, and
has arms of equal length and pans of equal weight. If
the arms were unequal, the rule would not be found to
be true unless it were carefully prescribed in which pan
the bodies were placed during the judgment of equality.
Again, the rules would not be true of the property length,
unless the rods were straight and were rigid. In implying
that the balance is good, and the rods straight and rigid,
we have implied definite laws which must be true if the
properties are to be measurable, namely that it is possible
to make a perfect balance, and that there are rods which
are straight and rigid. These are experimental laws ;
they could not be known apart from definite experiment
and observation of the external world ; they are not self-
t evident.

Accordingly the process of discovering that a property

is measurable in the way that has been described, and

/ setting up a process for measuring it, is one that rests

entirely upon experimental inquiry. It is a part, and a

120 WHAT IS SCIENCE?

most important part, of experimental science. Whenever
a new branch of physics is opened up (for, as has been
said, physics is the science that deals with such processes of
measurement), the first step is always to find some process
for measuring the new properties that are investigated ;
and it is not until this problem has been solved,
that any great progress can be made along the branch.
Its solution demands the discovery of new laws. We
can actually trace the development of new measurable
properties in this way in the history of science. Before
the dawn of definite history, laws had been discovered
which made measurable some of the properties employed
by modern science. History practically begins with
the Greeks, but before their time the properties, weight,
length, volume, and area had been found to be measur-
able ; the establishment of the necessary laws had prob-
ably occurred in the great period of Babylonian and
and Egyptian civilization. The Greeks, largely in the
person of Archimedes, found how to measure force by
establishing the laws of the lever, and other mechanical
systems. Again from the earliest era, there have been
rough methods of measuring periods of time,1 but a true
method, really obeying the three rules, was not discovered
till the seventeenth century ; it arose out of Galileo's
laws of the pendulum. Modern science has added greatly
to the list of measurable properties ; the science of elec-
tricity is based on the discoveiy, by Cavendish and
Coulomb, of the law necessary to measure an electric
charge ; on the laws, discovered by (Ersted and Ampere,
necessary to measure an electric current ; and on the
laws, discovered by Ohm and Kirchhoff, necessary to

1 By a period of time I mean the thing that is measured when we
say that it took us 3 hours to do so-and-so. This is a different
" time " from that which is measured when we say it is 3 o'clock.
The difference is rather abstruse and cannot be discussed here ; but
it may be mentioned that the " measurement " involved in "3
o'clock " is more like that discussed later in the chapter.

MEASUREMENT 121

measure electrical resistance. And the discovery of
similar laws has made possible the development of other
branches of physics.

But, it may be asked, has there ever been a failure to
discover the necessary laws ? The answer is that there
are certainly many properties which are not measurable
in the sense that we have been discussing ; there are more
properties, definitely recognized by science, that are not
so measurable than are so measurable. But, as will
appear presently, the very nature of these properties
makes it impossible that they should be measured in this
way. For the only properties to which this kind of
measurement seems conceivably applicable, are those
which fulfil the condition stated provisionally on p. in ;
they must be such that the combination of objects
possessing the property increases that property. For
this is the fundamental significance of the property
number ; it is something that is increased by addition ;
any property which does not agree with number in this
matter cannot be very closely related to number and
cannot possibly be measured by the scheme that has been
described. But it will be seen that fulfilment of this
condition only makes rule (2) true ; it is at least conceiv-
able that a property might obey rule (2) and not rules (i)
and (3). Does that ever happen, or can we always find
methods of addition and of judging equality such that, if
rule (2) is true, the laws are such that rules (i) and (3)
are also true ? In the vast majority of cases we can find
such methods and such laws ; and it is a very remarkable
fact that we can ; it is only one more instance of the way
in which nature kindly falls in with our ideas of what
ought to be. But I think there is one case in which the
necessary methods and laws have not yet been found and
are not likely to be found. It is a very difficult matter
concerning which even expert physicists might differ,
and so no discussion of it can be entered on here. But

122 WHAT IS SCIENCE?

it is mentioned in order to impress the reader with the
fact that measurement does depend upon experimental
laws ; that it does depend upon the facts of the external
world ; and that it is not wholly within our power to
determine whether we will or will not measure a certain
property. That is the feature of measurement which it
is really important to grasp for a proper understanding of
science.

MULTIPLICATION

Before we pass to another kind of measurement refer-
ence must be made to a matter which space does not
allow to be discussed completely. In stating the rules
that were necessary in order that weight should be measur-
able (p. 117), it was said that a collection having the same
weight as any given body could be made by adding other
bodies to that first selected. Now this statement is
not strictly true ; it is only true if the body first selected
has a smaller weight than any other body it is desired to
weigh ; and even if this condition is fulfilled, it is not true
if the bodies added successively to the collection are oi
the same weight as that first selected. Thus if my first
body weighs i lb., I cannot by adding to it make a collec-
tion which weighs less than i lb., and by adding bodies
which each weigh i lb., I cannot make a collection which
has the same weight as a body weighing (say) 2 J lb.

These facts, to which there is no true analogy in con-
nexion with number, force us to recognize " fractions."
A considerable complication is thereby introduced, and
the reader must accept my assurance that they can all
be solved by simple developments of the process of
measurement that has been sketched. But for a future
purpose it is necessary to notice very briefly the processes
of the multiplication and division of magnitudes on which
the significance of fractions depends.

MEASUREMENT 123

Suppose I have a collection of bodies, each of .which has
the same weight 3, the number of bodies in the collection
being 4. I may ask what is the weight of the whole
collection. The answer is given of course by multiplying
3 by 4, and we all know now that the result of that opera-
tion is 12. That fact, and all the other facts summed up
in the multiplication table which we learn at school, can
be proved from the rules on which weighing depend
together with facts determined by counting numerals.
But the point I want to make is that multiplication repre-
sents a definite experimental operation, namely the com-
bination into a single collection, placed on one pan of the
balance, of a set of bodies, all of the same weight, the
number of those bodies being known. Division arises
directly out of multiplication. In place of asking what
will be the weight of a collection formed of a given number
of bodies all of the same weight, we ask what must be the
weight of each of a collection of bodies, having a given
number, when the whole collection has a given weight.
E.g. what must each body weigh in order that the whole
collection of 4 bodies weighs 12 ? The answer is obtained
by dividing 12 by 4. That answer is obtained, partly
from the multiplication table, partly by inventing new
numerals which we call fractions ; but once again division
corresponds to a definite experimental operation and has
its primary significance because it corresponds to that
operation. This is this conclusion that we shall use in
the sequel. But it te worth while noting that the fractions
which we obtain by this method of addition overcome
the difficulty from which this paragraph started. If
we make all possible fractions of our original weight (i.e.
all possible bodies, such that some number of them formed
into a single collection have the same weight as the original
body), then, by adding together suitable collections of
these fractions, we can make up a collection which will
have the same weight as any body whatever that we

124 WHAT IS SCIENCE?

desire to weigh. This result is an experimental fact
which could not have been predicted without experi-
mental inquiry. And the result is true, not only for the
measurable property weight, but for all properties measur-
able by the process that is applicable to weight. Once
more we see how much simpler and more conveniently
things turn out than we have really any right to expect ;
measurement would have been a much more complex
business if the law that has just been stated were not
always true.

DERIVED MEASUREMENT

Measurement, it was said on p. no, is the assignment
of numbers (or, as we say now, numerals) to represent
properties. We have now considered one way in which
this assignment is made, and have brought to light the
laws which must be true if this way is to be possible.
And it is the fundamental way. We are now going to
consider some other ways in which numerals are assigned
to represent properties ; but it is important to insist at
the outset, and to remember throughout; that these
other ways are wholly dependent upon the fundamental
process, which we have just been discussing, and must
be so dependent if the numerals are to represent " real pro-
perties " and to tell us something scientifically significant
about the bodies to which they are attached. This
statement is confirmed by history ; all properties mea-
sured in the definitely pre-scientific era were measured
(or at least measurable) by the fundamental process ;
that is true of weight, length, volume, area and periods
of time. The dependent measurement, which we are
now about to consider, is a product of definitely and con-
sciously scientific investigation, although the actual
discovery may, in a few cases, be lost in the mists of the
past.

MEASUREMENT 125

The property which we shall take as an example of this
dependent or, as it will be termed, derived measurement,
is density. Every one has some idea of what density
means and realizes, vaguely at least, why we say that
iron is denser than wood or mercury than water ; and
most people probably know how density is measured,
and what is meant when it is said that the density of
iron is 8 times that of wood, and the density of mercury
13! times that of water. But they will feel also that there
is something more scientific and less purely common-
sense about the measurement of density than about the
measurement of weight ; as a matter of fact the discovery
of the measurement of density certainly falls within the
historic period and probably may be attributed to
Archimedes (about 250 B.C.). And a little reflection will
convince them that there is something essentially different
in the two processes.

For what we mean when we say a body has a weight 2
is that a body of the same weight can be made by com-
bining 2 bodies of the weight i ; that is the fundamental
meaning of weight ; it is what makes weight physically
important and, as we have just seen, makes it measurable.
But when we say that mercury has a density 13 J we do
not mean that a body of the same density can be prepared
by combining 13 J bodies of the density i (water). For,
if we did mean that, the statement would not be true.
However many pieces of water we take, all of the same
density, we cannot produce a body with any different
density. Combine water with water as we will, the result-
ing body has the density of water. And this, a little
reflection will show, is part of the fundamental meaning
of density ; density is something that is characteristic
of all pieces of water, large and small. The density of
water, a " quality " of it, is something fundamentally
independent of and in contrast with the weight of water,
the " quantity " of it.

126 WHAT IS SCIENCE?

But the feature of density, from which it derives its
importance, makes it totally impossible to measure
density by the fundamental process discussed earlier in
the chapter. How then do we measure it ? Before we
answer that question, it will be well to put another. As
was insisted before, if measurement is really to mean
anything, there must be some important resemblance
between the property measured, on the one hand, and
the numerals assigned to represent it, on the other.
In fundamental measurement, this resemblance (or the
most important part of it) arises from the fact that the
property is susceptible to addition following the same
rules as that of number, with which numerals are so
closely associated. That resemblance fails here. What
resemblance is left ?

MEASUREMENT AND ORDER

There is left a resemblance in respect of " order."
The numerals are characterized, in virtue of their use
to represent numbers, by a definite order ; they are
conventionally arranged in a series in which the sequence
is determined : " 2 " follows " i " and is before " 3 " ;
" 3 " follows " 2 " and is before " 4 " and so on. This
characteristic order of numerals is applied usefully
for many purposes in modern life ; we " number "
the pages of a book or the houses of a street, not in
order to know the number of pages in the book or of
houses in the street — nobody but the printer or the
rate-surveyor cares about that — but in order to be able
to find any given page or house easily. If we want p. 201
and the book opens casually at p. 153 we know in which
direction to turn the pages.1 Order then is characteristic

1 Numerals are also used to represent objects, such as soldiers or
telephones, which have no natural order. They are used here because
they provide an inexhaustible series of names, in virtue of the ingenious
device by which new numerals can always be invented when the old
ones have been used up.

MEASUREMENT 127

of numerals ; it is also characteristic of the properties
represented by numerals in the manner we are
considering now. This is our feature which makes the
" measurement " significant. Thus, in our example, bodies
have a natural order of density which is independent
of actual measurement. We might define the words
" denser " or " less dense " as applied to liquids (and the
definition could easily be extended to solids) by saying
that the liquid A is denser than B, and B less dense than
A, if a substance can be found which will float in A but
not in B. And, if we made the attempt, we should find
that by use of this definition we could place all liquids
in a definite order, such that each member of the series
was denser than the preceding and less dense than the
following member. We might then assign to the first
liquid the density i, to the second 2, and so on ; and we
should then have assigned numerals in a way which would
be physically significant and indicate definite physical
facts. The fact that A was represented by 2 and B by 7
would mean that there was some solid body which would
float in B, but not in A. We should have achieved some-
thing that might fairly be called measurement.

Here again it is important to notice that the possibility
of such measurement depends upon definite laws ; we
could not have predicted beforehand that such an arrange-
ment of liquids was possible unless we knew these laws.
One law involved is this : If B is denser than A, and C
denser than B, then C is denser than A. That sounds
like a truism ; but it is not. According to our definition
it implies that the following statement is always true :
If a body X floats in B and sinks in A, then if another body
Y sinks in B it will also sink in A. That is a statement
of facts ; nothing but experiment could prove that it is
true ; it is a law. And if it were not true, we could not
arrange liquids naturally in a definite order. For the
test with X would prove that B was denser than A,

128 WHAT IS SCIENCE?

while the test with Y (floating in A, but sinking in B)
would prove that A was denser than B. Are we then to
put A before or after B in the order of density ? We
should not know. The order would be indeterminate
and, whether we assigned a higher or a lower numeral to
A than to B, the assignment would represent no definite
physical fact : it would be arbitrary.

In order to show that the difficulty might occur, and
that it is an experimental law that it does not occur, an
instance in which a similar difficulty has actually occurred
may be quoted. An attempt has been made to define
the " hardness " of a body by saying that A is harder
than B if A will scratch B. Thus diamond will scratch
glass, glass iron, iron lead, lead chalk, and chalk butter ;
so that the definition leads to the order of hardness :
diamond, glass, iron, lead, chalk, butter. But if there is
to be a definite order, it must be true in all cases that if
A is harder than B and B than C, then A is harder than
C ; in other words, if A will scratch B and B C, then A
will scratch C. But it is found experimentally that there
are exceptions to this rule, when we try to include all
bodies within it and not only such simple examples as
have been quoted. Accordingly the definition does not
lead to a definite order of hardness and does not
permit the measurement of hardness.

There are other laws of the same kind that have to be
true if the order is to be definite and the measurement
significant ; but they will not be given in detail. One
of them the reader may discover for himself, if he will
consider the propert}^ colour. Colour is not a property
measurable in the way we are considering, and for this
reason. If we take all reds (say) of a given shade, we
can arrange them definitely in an order of lightness and
darkness ; but no colour other than red will fall in this
order. On the other hand, we might possibly take all
shades and arrange them in order of redness — pure red,

MEASUREMENT 129

orange, yellow, and so on ; but in this order there would
be no room for reds of different lightness. Colours
cannot be arranged in a single order, and it is for this
reason that colour is not measurable as is density.

NUMERICAL LAWS

But though arrangement in this manner in an order
and the assignment of numerals in the order of the pro-
perties are to some extent measurement and represent
something physically significant, there is still a large
arbitrary element involved. If the properties A, B, C,
D, are naturally arranged in that order, then in assigning
numerals to represent the properties I must not assign
to A 10, to B 3, to C 25, to D 18 ; for if I did so the order
of the numerals would not be that of the properties.
But I have an endless number of alternatives left ; I
might put A i, B 2, C 3, D 4,; or A 10, B 100, C 1,000,
D 10,000 ; or A 3, B 9, C 27, D 81 ; and so on. In the
true and fundamental measurement of the first part of
the chapter there was no such latitude. When I had
fixed the numeral to be assigned to one property, there
was no choice at all of the numerals to be assigned to the
others ; they were all fixed. Can I remove this latitude
here too and find a way of fixing definitely what numeral
is to be assigned to represent each property ?

In some cases, I can ; and one of these cases is density.
The procedure is this. I find that by combining the
numerals representing other properties of the bodies, which
can be measured definitely according to the fundamental
process, I can obtain' a numeral for each body, and that
these numerals lie in the order of the property I want to
measure. % If I take these numerals as representing the
property, then I still get numerals in the right order, but
the numeral for each property is definitely fixed. An
example will be clearer than this general statement. In

180 WHAT IS SCIENCE?

the case of density, I find that if I measure the weight
and the volume of a body (both measurable by the funda-
mental process and therefore definitely fixed), and I divide
the weight by the volume, then the numerals thus
obtained for different bodies lie in the order of their den-
sities, as density was defined on p. 127. Thus I find that
I gallon of water weighs 10 lb., but I gallon of mercury
weighs 135 lb. ; the weight divided by the volume for
water is 10, for mercury is 135 ; 135 is greater than 10 ;
accordingly, if the method is correct, mercury should be
denser than water and any body which sinks in mercury
should sink in water. And that is actually found to be
true. If therefore I take as the measure of the density
of a substance, its weight divided by its volume, then I
get a number which is definitely fixed,1 and the order of
which represents the order of density. I have arrived
at a method of measurement which is as definitely fixed
as the fundamental process and yet conveys adequately
the physically significant facts about order.

The invention of this process of measurement for
properties not suited for fundamental measurement is a
very notable achievement of deliberate scientific investi-
gation. The process was not invented by common
sense ; it was certainly invented in the historic period,
but it was not until the middle of the eighteenth century
that its use became widespread.2 To-day it is one of the
most powerful weapons of scientific investigation ; and
it is because so many of the properties of importance to

1 Except in so far as I may change the units in which I measure
weights and volume. I should get a different number if I measured
the volume in pints and the weight in tons. But this latitude in the
choice of units introduces a complication which it will be better to
leave out of account here. There is no reason why we should not
agree once and for all to use the same units ; and if we did that the
complication would not arise.

2 I think that until the eighteenth century only two properties were
measured in this way which were not measurable by the fundamental
process, namely density and constant acceleration.

MEASUREMENT 131

other sciences are measured in this way that physics,
the science to which this process belongs, is so largely
the basis of other sciences. But it may appear exceed-
ingly obvious to the reader, and he may wonder why the
invention was delayed so long. He may say that the
notion of density, in the sense that a given volume of the
denser substance weighs more than the same volume of
the less dense, is the fundamental notion ; it is what we
mean when we speak of one substance being denser (or
in popular language " heavier ") than another ; and that
all that has been discovered in this instance is that the
denser body, in this sense, is also denser in the sense of
p. 127. This in itself would be a very noteworthy dis-
covery, but the reader who raises such an objection has
overlooked a yet more noteworthy discovery that is
involved.

For we have observed that it is one of the most charac-
teristic features of density that it is the same for all
bodies, large and small, made of the same substance. It
is this feature which makes it impossible to measure it
by the fundamental process. The new process will be
satisfactory only if it preserves this feature. If we are
going to represent density by the weight divided by the
volume, the density of all bodies made of the same sub-
stance will be the same, as it should be, only if for all of
them the weight divided by the density is the same, that
is to say, in rather more technical language, if the weight
is proportional to the density. In adopting the new pro-
cess for measuring density and assigning numerals to
represent it in a significant manner, we are, in fact, assum-
ing that, for portions of the same substance, whether they
are large or small, the weight is proportional to the
volume. If we take a larger portion of the same sub-
stance and thereby double the weight, we must find, if
the process of measurement is to be a success, that we also
double the volume]; and this law must be true for 'all

132 WHAT IS SCIENCE?

substances to which the conception of density is applicable
at all.

Of course every one knows that this relation is actually
true ; it is so familiar that we are apt to forget that it is
an experimental truth that was discovered relatively
late in the history of civilization, which easily might not
be true. Perhaps it is difficult to-day to conceive that
when we take " more " of a substance (meaning thereby
a greater volume) the weight should not increase, but it
is quite easy to conceive that the weight should not
increase proportionally to the volume ; and yet it is upon
strict proportionality that the measurement of density
actually depends. If the weight had not been propor-
tional to the volume, it might still have been possible to
measure density, so long as there was some fixed numerical
relation between weight and volume. It is this idea
of a fixed numerical relation, or, as we shall call it
henceforward, a numerical law, that is the basis of the
" derived " process of measurement that we are consider-
ing ; and the process is of such importance to science
because it is so intimately connected with such numerical
laws. The recognition of such laws is the foundation of
modern physics.

THE IMPORTANCE OF MEASUREMENT

For why is the process of measurement of such vital
importance ; why are we so concerned to assign numerals
to represent properties. One reason doubtless is that
such assignment enables us to distinguish easily and
minutely between different but similar properties. It
enables us to distinguish between the density of lead and
iron far more simply and accurately than we could
do by saying that lead is rather denser than iron, but not
nearly so dense as gold — and so on. But for that purpose

MEASUREMENT 133

the " arbitrary " measurement of density, depending
simply on the arrangements of the substances in their
order (p. 127), would serve equally well. The true
answer to our question is seen by remembering the con-
clusion, at which we arrived in Chapter III, that \lie terms
between which laws express relationships are themselves
based on laws and represent collections of other terms
related by laws. When we measure a property, either
by the fundamental process or by the derived process,
the numeral which we assign to represent it is assigned as
the result of experimental laws ; the assignment implies
laws. And therefore, in accordance with our principle,
we should expect to find that other laws could be dis-
covered relating the numerals so assigned to each other
or to something else ; while if we assigned numerals
arbitrarily without reference to laws and implying no
laws, then we should not find other laws involving these
numerals. This expectation is abundantly fulfilled, and
nowhere is there a clearer example of the fact that the
terms involved in laws themselves imply laws. When we
can measure a property truly, as we can volume (by the
fundamental process) or density (by the derived process)
then we are always able to find laws in which these pro-
perties are involved ; we find, e.g., the law that volume is
proportional to weight or that density determines, in a
certain precise fashion, the sinking or floating of bodies.
But when we cannot measure it truly, then we do not find
a law. An example is provided by the property " hard-
ness " (p. 128) ; the difficulties met with in arranging
bodies in order of hardness have been overcome ; but
we still do not know of any way of measuring, by the
derived process, the property hardness ; we know of no
numerical law which leads to a numeral which always
follows the order of hardness. And so, as we expect, we
do not know any accurate and general laws relating hard-
ness to other properties. It is because true measurement

134 WHAT IS SCIENCE?

is essential to the discovery of laws that it is of such vital
importance to science.

One final remark should be made before we pass on.
In this chapter there has been much insistence on the
distinction between fundamental measurement (such as
is applicable to weight) and derived measurement (such
as is applicable to density). And the distinction is
supremely important, because it is the first kind of
measurement which alone makes the second possible.
But the reader who, when he studies some science in detail,
tries, as he should, to discover which of the two processes
is involved in the measurement of the various properties
characteristic of that science, may occasionally find
difficulty in answering the question. It should be pointed
out, therefore, that it is quite possible for a property to
be measurable by both processes. For all properties
measurable by the fundamental process must have a
definite order; for the physical property, number, to
which they are so similar, has an order — the order of
" more " or " less." This order of number is reflected
in the order of the numerals used to represent number.
But if it is to be measurable by the derived process, it
must also be such that it is also a " constant " in a numeri-
cal law — a term that is just going to be explained in the
next chapter. There is nothing in the nature of funda-
mental measurement to show that a property to which it
is applicable may not fulfil this condition also ; and some-
times the condition is fulfilled, and then the property is
measurable either by the fundamental or the derived
process. However, it must be remembered that the pro-
perties involved in the numerical law must be such that
they are fundamentally measurable ; for otherwise the
law could not be established. The neglect of this condi-
tion is apt to lead to confusion ; but with this bare hint
the matter must be left.

CHAPTER VII

NUMERICAL LAWS AND THE USE OF
MATHEMATICS IN SCIENCE

NUMERICAL LAWS

IN the previous chapter we concluded that density
was a measurable property because there is a
fixed numerical relation, asserted by a " numerical
law," between the weight of a substance and its volume.
In this chapter we shall examine more closely the idea
of a numerical law, and discover how it leads to such
exceedingly important developments.

Let us first ask exactly what we do when we are trying
to discover a numerical law, such as that between weight
and volume. We take various portions of a substance,
measure their weights and their volumes, and put down
the result in two parallel columns in our notebook. Thus
I may find these results :

TABLE I

WEIGHX __ _ > voma/^fi WEIGHT VOLUME

7

2 14

3 21

4 28

10 70

29 203

I now try to find some fixed relation between the corres-
ponding numbers in the two columns ; and I shall succeed
in that attempt if I can find some rule whereby, starting
with the number in one column, I can arrive at the corres-
ponding number in the other. If I find such a rule — •
and if the rule holds good for all the further measure-
ments that I may make — then I have discovered a
numerical law.

In the example we have taken the rule is easy to find.

135

136 WHAT IS SCIENCE?

I have only to divide the numbers in the second column
by 7 in order to arrive at those in the first, or multiply
those in the first by 7 in order to arrive at those in the
second. That is a definite rule which I can always apply
whatever the numbers are ; it is a rule which might always
be true, but need not always be true ; whether or no it
is true is a matter for experiment to decide. So much is
obvious ; but now I want to ask a further and important
question. How did we ever come to discover this rule ;
what suggested to us to try division or multiplication
by 7 : and what is the precise significance of division and
multiplication in this connexion ?

THE SOURCE OF NUMERICAL RELATIONS

The answer to the first part of this question is given by
the discussion on p. 123. Division and multiplication
are operations of importance in the counting of objects ;
in such counting the relation between 21, 7, 3 (the third
of which results from the division of the first by the
second) corresponds to a definite relation between the
things counted ; it implies that if I divide the 21 objects
into 7 groups, each containing the same number of objects,
then the number of objects in each of the 7 groups is 3.
By examining such relations through the experimental
process of counting we arrive at the multiplication (or
division) table. This table, when it is completed, states
a long series of relations between numerals, each of which
corresponds to an experimental fact ; the numerals
represent physical properties (numbers) and in any given
relation (e.g. 7 x 3 = 21) each numeral represents a
different property. But when we have got the multipli-
cation table, a statement of relations between numerals,
we can regard it, and do usually regard it, simply as a
statement of relations between numerals ; we can think
about it without any regard to what those numerals

NUMERICAL LAWS AND MATHEMATICS 137

represented when we were drawing up the table. And if
any other numerals are presented to our notice, it is
possible and legitimate to ask whether these numerals,
whatever they may represent, are in fact related as are
the numerals in the multiplication table. In particular,
when we are seeking a numerical relation between the
columns of Table I, we may inquire, and it is natural
for us to inquire, whether by means of the multiplication
we can find a rule which will enable us to arrive at the
numeral in the second column starting from that in the
first.

That explains why it is so natural to us to try division
when we are seeking a relation between numbers. But
it does not answer the second part of the question ; for
in the numerical law that we are considering, the relation
between the things represented by the numerals is not
that which we have just noticed between things counted.
When we say that, by dividing the volume by 7, we can
arrive at the weight, we do not mean that the weight is
the volume of each of the things at which we arrive by
dividing the substance into 7 portions, each having the
same volume. For a weight can never be a volume, any
more than a soldier can be a number ; it can only be repre-
sented by the same numeral as a volume, as a soldier
can be represented by a numeral which also represents a
number.

The distinction is rather subtle, but if the reader is to
understand what follows, he must grasp it. The relation
which we have found between weight and volume is a
pure numerical relation ; it is suggested by the relation
between actual things, namely collections which we
count ; but it is not that relation. The difference may be
expressed again by means of the distinction between
numbers and numerals. The relation between actual
things counted is a relation between the numbers — which
are physical properties — of those things ; the relation

138 WHAT IS SCIENCE?

between weight and volume is a relation between numerals,
the numerals that are used to represent those properties.
The physical relation in the second case is not between
numbers at all, but between weight and volume which
are properties quite different from numbers ; it appears
very similar to that between numbers only because we
use numerals, originally invented to represent numbers,
to represent other properties. The relation stated by a
numerical law is a relation between numerals, and only
between numerals, though the idea that there may be
such a relation has been suggested to us by the study of
the physical property, number.

If we understand this, we shall see what a very remark-
able thing it is that there should be numerical laws at all,
and shall see why the idea of such a law arose compara-
tively late in the history of science. For even when we
know the relations between numbers, there is no reason
to believe that there must be any relations of the same
kind between the numerals which are used to represent,
not only numbers, but also other properties. Until we
actually tried, there was no reason to think that it must
be possible to find at all numerical laws, stating numerical
relations such as those of division and multiplication.
The fact that there are such relations is a new fact, and
ought to be surprising. As has been said so often, it
does frequently turn out that suggestions made simply
by our habits of mind are actually true ; and it is because
they are so often true that science is interesting. But
every time they are true there is reason for wonder and
astonishment.

And there is a further consequence yet more deserving
of our attention at present If we realize that the
numerical relations in numerical laws, though suggested
by relations between numbers, are not those relations,
we shall be prepared to find also numerical relations which
are not even suggested by relations between numbers,

NUMERICAL LAWS AND MATHEMATICS 139

but only by relations between numerals. Let me take
an example. Consider the pairs of numerals (i, i), (2, 4),
(3, 9), (4, 16) ... Our present familiarity with
numerals enables us to see at once what is the relation
between each pair ; it is that the second numeral of the
pair is arrived at by multiplying the first numeral by
itself ; i is equal to i x i, 4 to 2 x 2, 9 to 3 x 3 ; and
so on. But, if the reader will consider the matter, he will
see that the multiplication of a number (the physical
property of an object) by itself does not correspond to
any simple relation between the things counted ; by the
mere examination of counted objects, we should never
be led to consider such an operation at all. It is suggested
to us only because we have drawn up our multiplication
table and have reached the idea of multiplying one
numeral by another, irrespective of what is represented
by that numeral. We know what is the result of
multiplying 3x3, when the two 3's represent different
numbers and the multiplication corresponds to a physical
operation on things counted ; it occurs to us that the multi-
plication of 3 by itself, when the two 3's represent the
same thing, although it does not correspond to a physical
relation, may yet correspond to the numerical relation
in a numerical law. And we find once more that this
suggestion turns out to be true ; there are numerical
laws in which this numerical relation is found. Thus if
we measure (i) the time during which a body starting
from rest has been falling (2) the distance through which
it has fallen during that time, we should get in our
notebook parallel columns like this :

TABLE II

TIME DISTANCE

1 .. I

2 .. 4

3-9

TIME DISTANCE

4 . . 16

5 •• 25

6 .. 36

140 WHAT IS SCIENCE?

The numerals in the second column are arrived at by multi-
plying those in the first by themselves ; in technical
language, the second column is the " square " of the first.
Another example. In place of dividing one column
by some fixed number in order to get the other, we may
use the multiplication table to divide some fixed number
(e.g. i) by that column. Then we should ge^the table

1 . . i-oo

2 . . 0-50

3 • • 0'33

4 • • 0-25

5 .. 0-20

and so on. Here, again, is a pure numerical operation
which does not correspond to any simple physical relation
upon numbers ; there is no collection simply related
to another collection in such a way that the number of
the first is equal to that obtained by dividing i by the
number of the second. (Indeed, as we have seen that
fractions have no application to number, and since this
rule must lead to fractions, there cannot be such a rela-
tion.) And yet once more we find that this numerical
relation does occur in a numerical law. If the first
column represented the pressure on a given amount of
gas, the second would represent the volume of that gas.

So far, all the relations we have considered were
derived directly from the multiplication table. But
an extension of the process that we are tracing leads to
relations which cannot be derived directly and thus
carries us further from the original suggestions indicated
by mere counting. Let us return to Table II, and
consider what would happen if we found for the numerals
in the second column values intermediate between those
given. Suppose we measured the distance first and
found 2, 3, 5, 6, 7, 8, 10, n, 12, 13, 14, 15 . . . ; what
does the rule lead us to expect for the corresponding
entries in the first column, the values of the time. The
answer will be given if in the multiplication table we

NUMERICAL LAWS AND MATHEMATICS 141

can find numerals which, when multiplied by themselves,
give 2, 3, 5 . . . But a search will reveal that there are
no such numerals. We can find numerals which, when
multiplied by themselves give very nearly 2, 3, 5 . . . ;
for instance, 1-41, 173, 2-24 give 1-9881, 2*9929, 5-0166,
and we could find numerals which would come even
closer to those desired. And that is really all we want,
for our measurements are never perfectly accurate, and
if we can get numerals which agree very nearly with
our rule, that is all that we can expect. But the search
for such numerals would be a very long and tedious
business ; it would involve our drawing up an enormously
complicated multiplication table, including not only
whole numbers but also fractions with many decimal
places. And so the question arises if we cannot find
some simpler rule for obtaining quickly the number
which multiplied by itself will come as close as we please
to 2, 3, 4 ... Well, we can ; the rule is given in every
textbook of arithmetic ; it need not be given here.
The point which interests us is that, just as the simple
multiplication of two numerals suggested a new process,
namely the multiplication of a numeral by itself, so this
new process suggests in its turn many other and more
complicated processes. To each of these new processes
corresponds a new rule for relating numerals and for
arriving at one starting from another ; and to each new
rule may correspond a numerical law. We thus get many
fresh forms of numerical law suggested, and some of
them will be found to represent actual experiments.

This process for extending arithmetical operations
beyond the simple division and multiplication from
which we start ; the consequent invention of new rules
for relating numerals and deriving one from another ;
and the study of the rules, when they are invented — all
this is a purely intellectual process. It does not depend
on experiment at all ; experiment enters only when we

142 WHAT IS SCIENCE?

inquire whether there is an actual experimental law
stating one of the invented numerical relations between
measured properties. The process is, in fact, part of
mathematics, not of experimental science ; and one of
the reasons why mathematics is useful to science is that
it suggests possible new forms for numerical laws. Of
course the examples that have been given are extremely
elementary, and the actual mathematics of to-day has
diverged very widely from such simple considerations ;
but the invention of such rules leads, logically if not
historically, to one of the great branches of modern
mathematics, the Theory of Functions. (When two
numbers are related as in our tables, they are technically
said to be " functions " of each other.) It has been
developed by mathematicians to satisfy their own
intellectual needs, their sense of logical neatness and of
form ; but though great tracts of it have no bearing
whatever upon experimental science, it still remains
remarkable how often relations developed by the mathe-
matician for his own purposes prove in the end to have
direct and immediate application to the experimental
facts of science.

NUMERICAL LAWS AND DERIVED MEASUREMENT

In this discussion there has been overlooked temporarily
the feature of numerical laws which, in the previous
chapter, we decided gave rise to their importance,
namely, that they made possible systems of derived
measurement. In the first law, taken as an example
(Table I), the rule by which the numerals in the second
column* were derived from those in the first involved a
numeral 7, which was not a member of those columns,
but an additional number applicable equally to all
members of the columns. This constant numeral,
characteristic of the rule asserted by the numerical law,

NUMERICAL LAWS AND MATHEMATICS 143

represented a property of the system investigated and
permitted a derived measurement of that system. But
in Table II, there is no such constant numeral ; the
rule for obtaining the second from the first column is
simply that the numerals in the first column are to be
multiplied by themselves ; no other numeral is involved.
But this simplicity is really misleading ; we should not,
except by a mere " fluke/' ever get such a table as Table
II as a result of our measurements. The reason is this.
Suppose that, in obtaining Table II, we have measured the
time in seconds and the distance fallen in feet ; and
that we now propose to write down the result of exactly
the same measurements, measuring the time in minutes
and the distances in yards. Then the numerals in the
first column, representing exactly the same observations,
would all be divided by 60 and those in the second would
all be divided by 3 ; the observation which was repre-
sented before by 60 in the first column would now be
represented by i ; and the number in the second column
represented before by 3 would now be represented by i.
If I now apply the rule to the two columns I shall find it
will not work ; the second is not the first multiplied by
itself. But there will be a new rule, as the reader may
see for himself ; it will be that the second column is the
same as the first, when the first is (i) multiplied by itself,
and (2) the result multiplied by 1,200. And if we
measured the time and the distance in some other units
(say hours and miles), we should again have to amend
our rule, but it would only differ from the former rule in
the substitution for 1,200 of some other numeral. If we
choose our units in yet a third way, we should get a third
rule, and this time the constant 'numeral might be i.
We should have exactly Table II ; but we should get that
table exactly only because we had chosen our units of
time and distance in a particular way.
These considerations arejquite general. Whatever the

144 WHAT IS SCIENCE?

numerical law, the rule involved in it will be changed by
changing the unit in which we measure the properties
represented by the two columns ; but the change will
only consist in the substitution of one constant numeral
for another. If we chance to choose the units in some
particular way, that constant numeral may turn out to
be i and so will disappear from sight. But it will always
be there. There must be associated with every numerical
law, involving a rule for arriving at the numerals in one
column from those in the other, some constant numeral
which is applicable to all members of the column alike.
And this constant may always, as in the case of density,
be the measure of some property to which derived
measurement is applicable. Every numerical law there-
fore— this is the conclusion to be enforced — may give
rise to a system of derived measurement ; and as a matter
of fact all important numerical laws do actually so give
rise.

CALCULATION

But though the establishment of system of derived
measurement is one' use of numerical laws, they have
also another use, which is even more important. They
permit calculation. This is an extremely important
conception which deserves our close attention.

Calculation is the process of combining two or more
numerical laws in such a way as to produce a third numeri-
cal law. The simplest form of it may be illustrated by the
following example. We know the following two laws which,
in rather different forms, have been quoted before :
(i) the weight of a given volume of any substance is
proportional to its density ; (2) the density of a gas is pro-
portional to the pressure upon it. From these two laws
we can deduce the third law : the weight of a given
volume of any gas is proportional to the pressure upon it.

NUMERICAL LAWS AND MATHEMATICS 145

That conclusion seems to follow directly without any need
for further experiments. Accordingly we appear to have
arrived at a fresh numerical law without adducing any
fresh experimental evidence. But is that possible ?
All our previous inquiry leads us to believe that laws,
whether numerical or other, can only be proved by experi-
mental inquiry and that the proof of a new law without
new experimental evidence is impossible. How are we
to reconcile the two conclusions ? When we have
answered that question we shall understand what is the
importance of calculation for science.

Let us first note that it is possible, without violating
the conclusions already reached, to deduce something
from a numerical law by a process of mere thought with-
out new experiment. For instance, from the law that
the density of iron is 7, I can deduce that a portion of it
which has a volume i will have a weight 7. But this
deduction is merely stating in new terms what was
asserted by the original law ; when I said that the density
of iron was 7, 1 meant (among other things) that a volume
i had a weight 7 ; if I had not meant that I should never
have asserted the law. The " deduction " is nothing
but a translation of the law (or of part of it), into different
language, and is of no greater scientific importance than
a translation from (say) English into" French. One kind
of translation, like the other, may have useful results,
but it is not the kind of useful result that is obtained
from calculation. Pure deduction never achieves any-
thing but this kind of translation ; it never leads to
anything new. But the calculation taken as an example
does lead to something new. Neither when I asserted
the first law, nor when I asserted the second did I mean
what is asserted by the third ; I might have asserted the
first without knowing the second and the second without
knowing the first (for I might have known what the density
of a gas was under different conditions without knowing

10

146 WHAT IS SCIENCE?

precisely how it is measured) ; and I might have
asserted either of them, without knowing the third. The
third law is not merely an expression in different words
of something known before ; it is a new addition to
knowledge.

But we have added to knowledge only because we have
introduced an assertion which was not contained in the
two original statements. The deduction depends on the
fact that if one thing (A) is proportional to another thing
(B) and if B is proportional to a third thing (C), then A is
proportional to C. This proposition was not contained
in the original statements. But, the reader may reply,
it was so contained, because it is involved in the very
meaning of " proportional " ; when we say that A is
proportional to B, we mean to imply the fact which has
just been stated. Now that is perfectly true if we are
thinking of the mathematical meaning of " propor-
tional," but it is not true if we are thinking of the
physical meaning. The proposition which we have really
used in making our deduction is this : If weight is pro-
portional (in the mathematical sense) to density, when
weight is varied by taking different substances, then it
is also proportional to density when weight is varied by
compressing more of the same substance into the same
volume. That is a statement which experiment alone
can prove, and it is because we have in fact assumed
that experimental statement that we have been able to
11 deduce " a new piece of experimental knowledge. It
is involved in the original statements only if, when it is
said that density is proportional to pressure, it is implied
that it has been ascertained by experiment that the law
of density is true, and that there is a constant density
of a gas, however compressed, given by dividing the
weight by the volume.

The conclusion I want to draw is this. When we appear
to arrive at new scientific knowledge by mere deduction

NUMERICAL LAWS AND MATHEMATICS 147

from previous knowledge, we are always assuming some
experimental fact which is not clearly involved in the
original statements. What we usually assume is that
some law is true in circumstances rather more general
than those we have considered hitherto. Of course the
assumption may be quite legitimate, for the great value
of laws is that they are applicable to circumstances more
general than those of the experiments on which they are
based ; but we can never be perfectly sure that it is
legitimate until we try. Calculation, then, when it
appears to add anything to our knowledge, is always
slightly precarious ; like theory, it suggests strongly
that some law may be true, rather than proves definitely
than some law must be true.

So far we have spoken of calculation as if it were merely
deduction ; we have not referred to the fact that calcula-
tion always involves a special type of deduction, namely
mathematical deduction. For there are, of course, forms
of deduction which are not mathematical. All argument
is based, or should be based, upon the logical processes
which are called deduction ; and most of us are prepared
to argue, however slight our mathematical attainments.
I do not propose to discuss here generally what are the
distinctive characteristics of mathematical argument ;
for an exposition of that matter the reader should turn
to works in which mathematicians expound their own
study. 1 I want only to consider why it is that this kind of
deduction has such a special significance for science.
And, stated briefly, the reason is this. The assumption,
mentioned in the last paragraph, which is introduced in
the process of deduction, is usually suggested by the form
of the deduction and by the ideas naturally associated
with it. (Thus, in the example we took, the assumption is
suggested by the proposition quoted about proportionality

1 E.g. "An Introduction to Mathematics," by Prof. Whitehead, in
the Home University Library.

148 WHAT IS SCIENCE?

which is the idea especially associated by the form
of the deduction). The assumptions thus suggested by
mathematical deduction are almost invariably found to
be actually true. It is this fact which gives to mathe-
matical deduction its special significance for science.

THE NEWTONIAN ASSUMPTION

Again an example is necessary and we will take one
which brings us close to the actual use of mathematics
in science. Let us return to Table II which gives the
relation between the time for which a body has fallen
and the distance through which it has fallen. The falling
body, like all moving bodies, has a " velocity." By the
velocity of a body we mean the distance that it moves in
a given time, and we measure the velocity by dividing
that distance by that time (as we measure density by
dividing the weight by the volume). But this way of
measuring velocity gives a definite result only when the
velocity is constant, that is to say, when the distance
travelled is proportional to the time and the distance
travelled in any given time is always the same (compare
what was said about density on p. 130). This condition
is not fulfilled in our example ; the distance fallen in the
first second is I, in the next 3, in the third 5, in the next 7
— and so on. We usually express that fact by saying
that the velocity increases as the body falls ; but we
ought really to ask ourselves whether there is such a
thing as velocity in this case and whether, therefore, the
statement can mean anything. For what is the velocity
of the body at the end of the 3rd second— i.e. at the time
called 3. We might say that it is to be found by taking
the distance travelled in the second before 3, which is 5,
or in the second after 3, which is 7, or in the second of
which the instant " 3 " is the middle (from 2 J to 3^), which
turns out to be 6. Or again we might say it is to be found

NUMERICAL LAWS AND MATHEMATICS 149

by taking half the distance travelled in the two seconds
of which " 3 " is the middle (from 2 to 4) which is again 6.
We get different values for the velocity according to
which of these alternatives we adopt. There are doubt-
less good reasons in this example for choosing the alterna-
tive 6, for two ways (and really many more than two ways,
all of them plausible) lead to the same result. But if
we took a more complicated relation between time and
distance than that of Table II, we should find that these
two ways gave different results, and that neither of them
were obviously more plausible than any alternative. Do
then we mean anything by velocity in such cases and,
if so, what do we mean ?

It is here that mathematics can help us. By simply
thinking about the matter Newton, the greatest of
mathematicians, devised a rule by which he suggested
that velocity might be measured in all such cases. 1 It is
a rule applicable to every kind of relation between time
and distance that actually occurs ; and it gives the
" plausible " result whenever that relation is so simple
that one rule is more plausible than another. Moreover
it is a very pretty and ingenious rule ; it is based on ideas
which are themselves attractive and in every way it
appeals to the aesthetic sense of the mathematician.
It enables us, when we know the relation between time
and distance, to measure uniquely and certainly the
velocity at every instant, in however complicated a way
the velocity may be changing. It is therefore strongly
suggested that we take as the velocity the value obtained
according to this rule.

But can there be any question whether we are right or
wrong to take that value ; can experiment show that we
ought to take one value rather than another ? Yes, it

1 This is part of the great mathematical achievement mentioned on
p. 100. I purposely refrain from giving the rule, not because it is
really hard to explain, but because I want to make clear that what
is important is to have some rule, not any particular rule,

150 WHAT IS SCIENCE?

can ; and in this way. When the velocity is constant
and we can measure it without ambiguity, then we can
establish laws between that velocity and certain properties
of the moving body. Thus, if we allow a moving steel ball
to impinge on a lead block, it will make a dent in it deter-
mined by its velocity ; and when we have established by
observations of this kind a relation between the velocity
and the size of the dent, we can obviously use the size of
the dent to measure the velocity. Suppose now our
falling body is a steel ball, and we allow it to impinge
on a lead block after falling through different distances ;
we shall find that its velocity, estimated by the size of
the dent, agrees exactly with the velocity estimated by
Newton's rule, and not with that estimated by any other
rule (so long, of course, as the other rule does not give the
same result as Newton's). That, I hope the reader will
agree, is a very definite proof that Newton's rule is right.
On this account only Newton's rule would be very
important, but it has a wider and much more important
application. So far we have expressed the rule as giving
the velocity at any instant when the relation between
time and distance is known ; but the problem might be
reversed. We might know the velocity at any instant
and want to find out how far the body has moved in any
given time. If the velocity were the same at all instants,
the problem would be easy ; the distance would be the
velocity multiplied by the time. But if it is not the same,
the right answer is by no means easy to obtain ; in fact
the only way of obtaining it is by the use of Newton's
rule. The form of that rule makes it easy to reverse it
and, instead of obtaining the velocity from the distance,
to obtain the distance from the velocity ; but until that
rule was given, the problem could not have been solved ;
it would have baffled the wisest philosophers of Greece.
Now this particular problem is not of any very great
importance, for it would be easier to measure by experi-

NUMERICAL LAWS AND MATHEMATICS 151

ment the distance moved than to measure the velocity
and calculate the distance. But there are closely
analogous cases — one of which we shall notice immedi-
ately— in which the position will be reversed. Let us
therefore ask what is the assumption which, in accord-
ance with the conclusion reached on p. 146, must be
introduced, if the solution of the problem is to give new
experimental knowledge.

We have seen that the problem could be solved easily
if the velocity were constant ; what we are asking, is
how it is to be solved if the velocity does not remain
constant. If we examined the rule by which the solution
is obtained, we should find that it involves the assump-
tion that the effect upon the distance travelled of a certain
velocity at a given instant of time is the same as it would
be if the body had at that instant the same constant
velocity. We know how far the body would travel at
that instant if the velocity were constant, and the assump-
tion tells us that it will travel at that instant the same
distance although the velocity is not constant. To obtain
the whole distance travelled in any given time, we have to
add up the distances travelled at the instants of which
that time is made up ; the reversed Newtonian rule gives
a simple and direct method for adding up these distances,
and thus solves the problem. It should be noted that
the assumption is one that cannot possibly be proved by
experiment ; we are assuming that something would
happen if things were not what they actually are ; and
experiment can only tell us about things as they are.
Accordingly calculation of this kind must, in all strictness,
always be confirmed by experiment before it is certain.
But as a matter of fact, the assumption is one of which
we are almost more certain than we are of any experiment.
It is characteristic, not only of the particular example
that we have been considering, but of the whole structure
of modern mathematical physics which has arisen out

152 WHAT IS SCIENCE?

of the work of Newton. We should never think it really
necessary to-day to confirm by experiment the results
of calculation based on that assumption ; indeed if
experiment and calculation did not agree, we should
always maintain that the former and not the latter was
wrong. But the assumption is there, and it is primarily
suggested by the aesthetic sense of the mathematician,
not dictated by the facts of the external world. Its
certainty is yet one more striking instance of the
conformity of the external world with our desires.

And now let us glance at an example in which such
calculation becomes of real importance. Let us take a
pendulum, consisting of a heavy bob at the end of a
pivoted rod, draw it aside and then let it swing. We
ask how it will swing, what positions the bob will occupy
at various times after it is started. Our calculation
proceeds from two known laws, (i) We know how the
force on the pendulum varies with its position. That we
can find out by actual experiments. We hang a weight
by a string over a pulley, attach the other end of the
string to the bob, and notice how far the bob is pulled
aside by various weights hanging at the end of the string.
We thus get a numerical law between the force and the
angle which the rod of the pendulum makes with the
vertical. (2) We know how a body will move under a
constant force. It will move in accordance with Table II,
the distance travelled being proportional to the
" square " of the time during which the force acts. Now
we introduce the Newtonian assumption. We know the
force in each position ; we know how it would move in
that position if the force on it were constant ; actually
it is not constant, but we assume that the motion will be
the same as it would be if, in that position, the force were
constant. With that assumption, the general Newtonian
rule (of which the application to velocity is only a special
instance) enables us to sum up the effects of the motions

NUMERICAL LAWS AND MATHEMATICS 153

in the different positions, and thus to arrive at the desired
relation between the time and the positions successively
occupied by the pendulum. The whole of the calculation
which plays so large a part in modern science is nothing
but an elaboration of that simple example.

MATHEMATICAL THEORIES

We have now examined two of the applications of
mathematics to science. Both of them depend on the
fact that relations which appeal to the sense of the
mathematician by their neatness and simplicity are
found to be important in the external world of experiment.
The relations between numerals which he suggests are
found to occur in numerical laws, and the assumptions
which are suggested by his arguments are found to be
true. We have finally to notice a yet more striking
example of the same fact, and one which is much more
difficult to explain to the layman.

This last application is in formulating theories. In
Chapter V we concluded that a theory, to be valuable,
must have two features. It must be such that laws
can be predicted from it and such that it explains these
laws by introducing some analogy based on laws more
familiar than those to be explained. In recent develop-
ments of physics, theories have been developed which
conform to the first of these conditions but not to the
second. In place of the analogy with familiar laws,
there appears the new principle of mathematical
simplicity. These theories explain the laws, as do the
older theories, by replacing less acceptable by more
acceptable ideas ; but the greater acceptability of the
ideas introduced by the theories is not derived from an
analogy with familiar laws, but simply from the strong
appeal they make to the mathematician's sense of
form,

154 WHAT IS SCIENCE?

I do not feel confident that I can explain the matter
further to those who have not some knowledge of both
physics and mathematics, but I must try. The laws
on the analogy with which theories of the older type are
based were often (in physics, usually) numerical laws,
such laws for example as that of the falling body. Now
numerical laws, since they involve mathematical relations,
are usually expressed, not in words, but in the symbols in
which, as every one knows, mathematicians express their
ideas and their arguments. I have been careful to avoid
these symbols; until this page there is hardly an "x"
or a " y " in the book. And I have done so because
experience shows that they frighten people ; they make
them think that something very difficult is involved.
But really, of course, symbols make things easier ; it is
conceivable that some super-human intellect might be
able to study mathematics, and even to advance it,
expressing all his thoughts in words. Actually, the
wonderful symbolism mathematics has invented make
such efforts unnecessary ; they make the processes
of reasoning quite easy to follow. They are actually
inseparable from mathematics ; they make exceedingly
difficult arguments easy to follow by means of simple
rules for juggling with these symbols — interchanging
their order, replacing one by another, and so on. The
consequence is that the expert mathematician has a
sense about symbols, as symbols ; he looks at a page
covered with what, to anyone else, are unintelligible
scrawls of ink, and he immediately realizes whether the
argument expressed by them is such as is likely to satisfy
his sense of form ; whether the argument will be " neat "
and the results " pretty." (I can't tell you what those
terms mean, any more than I can tell you what I mean
when I say that a picture is beautiful.)

Now sometimes, but not always, simple folk can
understand what he means ; let me try an example.

NUMERICAL LAWS AND MATHEMATICS 155

Suppose you found a page with the following marks on
it — never mind if they mean anything :

dy dp dX dy dp

dy dz dt dy dz

da dy dY da dy

!=__»- — — — _ __

dz dx dt dz dx

dp da dZ dp da

dx dy dt dx dy

da dY dZ da dY dZ

dt dz dy dt dz dy

dp dZ dX dp dZ dX

dt dx dz dt dx dz

dy dX dY dy dX dY

dt dy dx dt dy dx

I think you would see that the set of symbols on the
right side are " prettier " in some sense than those on
the left ; they are more symmetrical. Well, the great
physicist, James Clerk Maxwell, about 1870, thought
so too ; and by substituting the symbols on the right
side for those on the left, he founded modern physics,
and, among other practical results, made wireless
telegraphy possible.

It sounds incredible ; and I must try to explain a little
more. The symbols on the left side represent two
well-known electrical laws : Ampere's Law and Faraday's
Law ; or rather a theory suggested by an analogy with
those laws. The symbols i, j, k represent in those laws

156 WHAT IS SCIENCE?

an electric current. For these symbols Maxwell sub-

dX dY dZ

stituted — — — ; that substitution was roughly
dt dt dt

equivalent to saying that an electric current was related
to the things represented by X, Y, Z, t (never mind what
they are) in a way nobody had ever thought of before ;
it was equivalent to saying that so long as X, Y, Z, t were
related in a certain way, there might be an electric
current in circumstances in which nobody had believed
that an electric current could flow. As a matter of
fact, such a current would be one flowing in an absolutely
empty space without any material conductor along
which it might flow, and such a current was previously
thought to be impossible. But Maxwell's feeling for
symbolism suggested to him that there might be such a
current, and when he worked out the consequences of
supposing that there were such currents (not currents
perceptible in the ordinary way, but theoretical currents,
as molecules are theoretical hard particles), he arrived
at the unexpected result that an alteration in an electric
current in one place would be reproduced at another
far distant from it by waves travelling from one to the
other through absolutely empty space between. Hertz
actually produced and detected such waves ; and
Marconi made them a commercial article.

That is the best attempt I can make at explaining the
matter. It is one more illustration of the marvellous
power of pure thought, aiming only at the satisfaction
of intellectual desires, to control the external world.
Since Maxwell's time, 'there have been many equally
wonderful theories, the form of which is suggested by
nothing but the mathematician's sense for symbols. The
latest are those of Sommerfeld, based the ideas of Niels
Bohr, and of Einstein. Every one has heard of the latter,
but the former (which concerns the constitution of the

NUMERICAL LAWS AND MATHEMATICS 157

atom) is quite as marvellous. But of these I could not give,
even if space allowed, even such an explanation as I
have attempted for Maxwell's. And the reason is this :
A theory by itself means nothing experimental — we
insisted on that in Chapter V — it is only when something
is deduced from it that it is brought within the range of
our material senses. Now in Maxwell's theory, the
symbols, in the alteration of which the characteristic
feature of the theory depends, are retained through the
deduction and appear in the law which is compared with
experiment. Accordingly it is possible to give some idea
of what these symbols mean in terms of things experi-
mentally observed. But in Sommerfeld's or Einstein's
theory the symbols, which are necessarily involved in the
assumption which differentiates their theories from
others, disappear during the deduction ; they leave a
mark on the other symbols which remain and alter the
relation between them ; but the symbols on the relations
of which the whole theory hangs, do not appear at all
in any law deduced from the theory. It is quite impos-
sible to give any idea of what they mean in terms of experi-
ment.1 Probably some of my readers will have read the
very interesting and ingenious attempts to " explain
Einstein " which have been published, and will feel that
they really have a grasp of the matter. Personally I
doubt it ; the only way to understand what Einstein did
is to look at the symbols in which his theory must
ultimately be expressed and to realize that it was reasons
of symbolic form, and such reasons alone, which led him
to arrange the symbols in the way he did and in no other.
But now I have waded into such deep water that it is
time to retrace my steps and return to the safe shore of
the affairs of practical life.

1 The same is true really of the exposition of the Newtonian
assumption attempted on p. 151. It is strictly impossible to state
exactly what is the assumption discussed there without using
symbols. The acute reader will have guessed already that on that
page I felt myself skating on very thin ice.

CHAPTER VIII
THE APPLICATIONS tF SCIENCE

THE PRACTICAL ^ VALUE •? SCIENCE

SO far we have regarded science as a means of satis-
fying our purely intellectual desires. And it must
be insisted once again that such is_thejmnaryjmd
fundamental object of science ; if science did not
fulfil that purpose, then it could certainly fulfil no
other. It has applications to practical life, only because
it is true ; and its truth arises directly and immediately
from its success as an instrument of intellectual satis-
faction. Nevertheless there is nt dtubt that, for the
world at large — the world which includes those to which
this book is addressed — it is the practical rather than the
intellectual value of science which makes the greater
appeal. I do not mean that they are blind to the things
of the mind, and consider only those of the body ; I mean
merely that science is not for them the most suitable
instrument by which they may cultivate their minds.
Art, history, and philosophy are competing vehicles of
culture ; and their sense of the supreme value of their
own study should not lead men of science to insist that
its value is unique. Indeed, if we are forced to recognize
that pure science will always be an esoteric study, it
should increase our pride that we are to be found in the
inner circle of the elect. On the other hand, since man
cannot live by thought alone, the practical value of science
makes a universal appeal ; it would be pedantic and
misleading to omit some consideration of this aspect of
science.

The practical value of science arises, of course, from the
formulation of laws. Laws predict the behaviour of that

158

THE APPLICATIONS 3F SCIENCE 159

external world with which our practical and everyday
life is an unceasing struggle. Forewarned is forearmed,
and we stand a better chance of success in the contest
if we know precisely how our adversary may be expected
to behave. Knowledge is power and our knowledge of
the external world enables us in some measure to control
it. So much is obvious ; nobody to-day will be found to
deny that science — and it must be remembered that we
always use that word to denote the abstract study —
might be of great service in practical life. Nor indeed
will anybody deny that it has been of great service. We
have all heard how the invention of the dynamo — on which
is based every industrial use of electricity, without which
modern civilization would be impossible — or the discovery
of the true nature of ferments — the basis of modern
medicine — was the direct outcome of the purest and most
disinterested intellectual inquiries. But although this
is granted universally, men of science are still heard to
complain with ever-increasing vehemence that they are
not allotted their due share of influence in the control of
industry and of the State, and that science is always
suffering from material starvation. It is clear, therefore,
that in spite of the superficial agreement on the value of
science, there is still an underlying difference of opinion
which merits our attention.

The difference is not surprising, for candour compels
us to confess that these admitted facts, on which the
claims of science to practical value are often based, are
not really an adequate basis for those claims. The fact
that science might produce valuable results and actually
has produced some, is no more justification for our
devoting any great part of our energies to its develop-
ment than the fact that I picked up half a crown in the
street yesterday — and might pick up another — would be a
justification for my abandoning sober work to search for
buried treasure. Moreover, the very people who claim,

160 WHAT IS SCIENCE?

on the ground of the work of Faraday or Pasteur, that
science should receive large endowments and a great
share in government, often urge at the same time that
Faraday and Pasteur were examples of that genius
which cannot be produced by training and can scarcely
be stunted by adversity. If it were only these exceptional
achievements, occurring two or three times a century,
which had practical value, the encouragement of science
would be an unprofitable gamble. If we are really to
convince the outside world of the need for the closer
application of science to practical affairs, we must give
reasons for our claim much more carefully and guardedly
than has been the custom up to the present. Nothing
is more fatal to our cause than to encourage expectations
doomed to disappointment.

Accordingly in this chapter, I propose to diverge
entirely from the usual path. I shall not give a single
example of practical science. There are plenty of good
books which tell what science has achieved in the past,
and plenty of newspaper paragraphs to tell us what it
is going to achieve in the future. Here I want to inquire
carefully what value science might have for practical
life, why it has that value, and under what conditions
its value is most likely to-be realized.

THE LIMITATIONS OF SCIENCE

It will be well to point out immediately that science,

like everything else, has its limitations, and that there are

some practical problems which, from its very nature, it

is debarred from solving. It must never be forgotten

'.that, though science helps us in controlling the external

1 world, it does not give us the smallest indication in what

I direction that control should be exercised. Whenever we

undertake any practical action, we have two decisions to

make ; we have to decide what is the end of our action,

THE APPLICATIONS OF SCIENCE 161

what result we wish to obtain ; and we have to decide
what is the right means to that end, what action will
produce the desired result. The distinction between the
two decisions can be traced in the simplest as well as
(perhaps better than) in the most complex actions. If I
go to a meal in a restaurant, I have first to decide
whether I want beef or mutton, tea or coffee, and
second how I am to get what I want. If I have tooth-
ache, I have first to decide whether I want to be cured,
and second if I am more likely to be cured if I doctor
myself or if I go to a dentist. The fact that there are
two decisions is sometimes obscured by the simplicity
and obviousness of one of them. In the first example,
the decision as to means is liable to be overlooked ; for
(except in some restaurants) it is obvious that the best
way to get the meal I want is to ask for it. In the second,
the decision as to end may be unobserved, because it is
so obvious that I want to be cured.

In these simple examples the distinction between the
two decisions is clear ; in others they are so closely inter-
connected that care is needed to separate them. Our
choice of the ends at which we may aim is often deter-
mined in part by the means we have of attaining them ;
it is foolish to struggle towards a goal that can never be
reached. On the other hand, action which is desirable
as a means to one end, may be objectionable because it
leads at the same time to other results that are undesirable
as ends. In all the more complicated decisions of life,
such conflicts between ends and means arise, and it is a
necessary step towards accuracy of thought to disentangle
the conflicting elements. It is all the more necessary/
because in controversial matters there is always a*
tendency to conceal questions of ends and to pretend',
that every question is one of taieans only ; the reason is >
that agreement concerning ends is far less j easily \
attainable than agreement concerning means, so that, *

162 WHAT IS SCIENCE?

when we are trying to make converts to our views, we
are naturally apt to disguise differences that are irrecon-
cilable.

Political discussion provides examples of this ten-
dency. It is clap-trap to announce portentously that
we all desire the welfare of the community and to pretend
that we differ only in our view of the best way of attain-
ing it ; what we really differ about is our ideas of the wel-
fare of the community ; we disagree as to what is the state
of society that forms the end of our political action. If
we could agree about that, our remaining differences
would not excite much heat. As it is, our pretence that
we are arguing merely about means often leads us to
adopt means which are obviously ill-adapted to secure
any of the ends at which any of the contending parties
aim.

Since science must always exclude from its province

•'judgments concerning which differences are irreconcilable,

[it can only guide practical life in the choice of means, and

1 not in the clwice of ends. If one course of action is more

' " scientific " than another, that course is better only in

the sense that is a more efficient means to some end ;

from the fact that it is indicated as a result of scientific

inquiry, it is quite illegitimate to conclude that the action

must necessarily be desirable. That conclusion follows

only if it can be proved that the end, to which the action

is a means, is desirable ; such a proof must always lie

wholly without the range of science. The neglect to

observe this distinction is responsible for much of the

disregard, and even actual dislike, of their study in its

application to practical life of which men of science so

often complain. It was seriously urged in recent years

that science, being responsible for the horrors of modern

warfare, is a danger to civilization ; and I am told that

many manual workers are inclined to regard science with

hostility because it is associated in their minds with the

THE APPLICATIONS OF SCIENCE 163

" scientific " management of industry.1 Such objections
are altogether unjust ; science gives to mankind a greater
power of control over his environment. He may use that
control for good or for ill ; and if he uses it for ill, the fault
lies in that part of human nature which is most remote
from science ; it lies in the free exercise of will. To deny
knowledge for fear it may be misused is to repeat the
errors of the mediaeval church ; thus to deprive men of
the power to do evil is to deprive them of the yet greater
power to do good. For precisely the same knowledge
that has made Europe a desert has given the power to
restore her former fertility ; and the increase of individual
productive power which may be used to rivet more closely
the chains of wage-slavery might also give to the worker
that leisure from material production which alone can
give freedom to the slave.

Men of science themselves are largely to blame for the
confusion against which this protest is directed. They
are so accustomed to having to force their conclusions on
an ignorant and reluctant world, that they are apt to
overstep the limits of their special sphere ; they some-
times forget that they cease to be experts when they
leave their laboratories, and that in deciding questions
foreign to science, they have no more (but, of course,
no less) claim to attention than anyone else. Like the
members of any other trade or profession, they are apt
to be affected in their social and political views by the
work which is their main occupation, and to lay special
stress on the evils which come immediately under their
notice.2 In this respect it is useless to expect them to

1 We need not discuss here whether the methods of factory control
to which objection is taken have really any claim to be scientific in our
sense ; whether, that is to say, they are the outcome of such investiga-
tion as has been described in the earlier chapters.

2 I am tempted to describe what are the social and political views
which the study and practice of science tends to inculcate. But this
is a matter in which the spectator sees most of the game ; if I made
the attempt, I should probably be led astray by my own particular
opinions.

164, WHAT IS SCIENCE?

be more perfect than the rest of mankind. But any
danger of paying too much or too little heed to pronounce-
ments put forward on behalf of " science " will be
avoided if the distinction on which so much stress has
been laid is borne in mind. On questions of means to a
given end (if they concern the nature of the external
world) science is the one and only true guide ; on ques-
tions of the ends to which means should be directed,
science has nothing to say.

THE CERTAINTY OF SCIENTIFIC KNOWLEDGE

I have thought it better thus to start with a considera-
tion of the limitations of science ; not because the greater
danger lies in the neglect of those limitations, but merely
to convince the reader that I am not blind to their exist-
ence. Actually, in this country at least, the greater
danger lies in the other direction, in refusing to accept
the clear and positive decisions of science on matters
which lie wholly within its bounds. Why is there any
such danger ? It arises, I believe, from two sources not
wholly independent. The first source is a disbelief lhat
science is really possessed of any definite knowledge.
Scientific experts seem to differ as much as experts in
other subjects, and may be heard in any patent litiga-
tion swearing cheerfully against each other. The second
source is a general distrust of the " theorist " as com-
pared with the " practical man." The chief points that
have to be raised will appear naturally in a discussion of
these two errors.

It may be thought that the first " error " has been
implicitly confirmed by our previous discussions. For it
has been urged that there is a strong personal element in
science and that complete agreement is to be found only
in its subject matter and not in its conclusions. But while
it is perfectly true that a theory, and even to some extent

THE APPLICATIONS OF SCIENCE 165

a law, may be an object of contention when it is first
proposed, it is equally true that the difference of opinion
is always ultimately resolved. A theory may be doubted,
but while it is doubted it is not part of the firm fabric
of science ; but in the end it is always either definitely
accepted or definitely rejected. It is in this that science
differs from such studies as history or philosophy in
which controversies are perennial. There is an immense
body of science concerning which there is no doubt,
and that body includes both theories and laws ; there is
a smaller part concerning which dispute is still continu-
ing. It is only natural that this smaller part should
receive the greater share of explicit attention ; the other
and greater part we learn in our school and university
courses and find no need to discuss later, because it is a
matter of common knowledge with which all properly
informed persons are completely familiar ; it is the base
from which we proceed to establish new knowledge, and
the premiss on which we found our arguments concern-
ing it. The distinction between the twojpajts

knowledge, that which is firmly established, and that
which is still doubtful, is perfectly clear and definite to
all who have been properly trained. The fact that doctors
differ in science, as in other things, does not affect the
equally important fact that in much the larger part of
their knowledge they agree.

But a more serious objection may be raised. In the
opening chapters we concluded that science draws its
subject-matter from a limited portion of experience, and
that this limited portion necessarily excludes all that part
of our life which is of the most intimate interest to us.
It may be urged with force that while science may be in
possession of perfectly positive knowledge concerning
which every one who has studied the matter is in agree-
ment, yet this knowledge is entirely divorced from all
the affairs of practical life ; when science attempts to

166 WHAT IS SCIENCE?

intrude into such affairs it becomes as hesitating and
dubi table as any other source of knowledge.

As a formal statement of the position, this objection
must be admitted as valid. The uniformly certain and
completely universal laws of science can be realized only
in the carefully guarded conditions of the laboratory, and
are never found in the busy world outside. There is
scarcely any event or process of practical importance to
which we could point as providing a direct confirmation
of any of the propositions of pure science, or which could
be described completely in terms of those propositions.
In every such event and process, there is involved some
element of which science can take no cognizance, and it
is usually on account of this element (as was remarked in
Chapter III) that the event has practical importance.
And again, it is the presence of this element which makes
it possible for experts, equally well-informed, to differ
in their preliminary suggestions of an explanation of the
event or of the most suitable means for controlling it.
But it does not follow because practical events do not
lie wholly within the realm of science that they lie .wholly
without it. Indeed it is from the study of practically
important events that many of the results of pure science
have actually been derived. Let us examine the matter
more carefully.

All the applications of science to practical life depend
ultimately on a knowledge of laws. Whether we are
asked to explain an event, or to suggest means whereby
an event may be produced or prevented, we can meet
the demand only if we know the laws of which the event
is the consequence. But laws state only relations between
events ; when we say that an event is the consequence
of certain laws, we do not mean that this event must
happen in all possible circumstances ; we mean only that
it is invariably associated with certain other events,
and must happen if they happen. The event in question

THE APPLICATIONS OF SCIENCE 167

is not only a consequence of the laws, it is a consequence
of the laws and of the other events to which it is related
by the laws. Again, it must be noted that I have spoken
of laws, not " a law/' The practical event to which our
attention is directed will not be a simple event such as
is related by pure science to another simple event ; it
will be an immensely complicated collection of such
events, and these constituent events will be each related
to some other event by a separate law. The consti-
tuent events will not in general be related to each other
by a law, nor will the other events, to which they are so
related, be related to each other by a law. The explana-
tion of the events in question will not be complete when
we have stated a single law, or even all the many laws
in which the constituent events are involved ; it is neces-
sary to add that the many events to which they are related
by these many laws have actually occurred, and that the
many laws are actually in operation.

The last part of the explanation is the part which is
not pure science. Science when it asserts laws, only
asserts that, if so-and-so happens, something else will
also happen. But in practical matters it is necessary to
convert this hypothetical statement into a definite state-
ment, and to assert that something actually has happened.
This is often an extremely difficult matter, which may be
the subject of much difference of opinion _ until all the
circumstances have been investigated. An obvious
example of this difficulty arises in the practice of
medicine ; diagnosis, the determination of what is wrong
with the patient, is a necessary preliminary to his treat-
ment, and is actually the gravest problem with which
the physician is faced. And similar problems arise in all
other branches of applied science. If we are asked to
produce some desired product or to find out why the
product of some existing process is not satisfactory, the
first part of our task must always be to find out exactly

168 WHAT IS SCIENCE?

in what the desired product consists, and exactly in
what particulars it differs from the unsatisfactory product.
This problem of determining precisely what are the exist-
ing facts is not strictly one of science at all ; the solution
of it does not involve the statement of any scientific laws,
for laws assert, not what does actually occur, but what
will occur if something else occurs. Nevertheless science
and scientific laws are useful, and even indispensable, in
the solving of it ; for very often the best or only proof
that something has occurred is that some other event
has occurred with which the first is associated by a law.
Thus, the physician bases his diagnosis on his examina-
tion of symptoms ; he observes that the bodily state of
his patient is abnormal in some particulars, and from his
knowledge of the laws connecting those parts of the body
which are accessible to observation with those that are
not deduces what must be the state of the hidden organs.
In the same way the works chemist or physicist is often
led to judge what is the source of failure in a product,
by examining carefully the process by which it has been
produced, and deducing by his knowledge of the laws
of chemistry or physics what must be the result of that
process.

It is for such reasons that pure science, although it
takes no direct cognizance of the actual events of practical
life, is of inestimable service in explaining and controlling
them. Even though it is impossible to analyse those
events completely into laws, it is only by carrying that
analysis as far as it will go, and by bringing to light all
the laws that are involved, that any explanation or
any control can be attained.

These considerations have been suggested by the first
of the two errors noted on page 164 ; they answer the
objection that science is not possessed of any positive
and certain knowledge, or that, if it has such knowledge, it
is not relevant to practical problems. The second

THE APPLICATIONS OF SCIENCE 169

error is even more dangerous. Science, it is often urged
(perhaps not in these actual words), is all very well ;
it may even be indispensable ; but it must be the right
kind of science. The kind of science that is needed in
everyday life is not that of the pure theorist, but that
which every practical man is bound to acquire for
himself in the ordinary conduct of his business.

Again, it will be well to begin by admitting that there
is some truth in the contention that the practical man
is likely to manage the business in which he has been
immersed all his life better than one who has no experience
of any conditions more complicated than those of the
laboratory. No doubt scientific men of great eminence
often prove as great failures in industry as commercial
men would in pure science. But we have already noticed
that no practical problem is wholly scientific ; there are
questions of ends as well as of means. The scientific
man in industry is doubtless apt to be led astray by
forgetting that the object of industry is to produce
goods, and that processes, however scientifically interest-
ing they may be, are commercially worthless unless they
decrease the expenditure of capital and labour necessary
to obtain a given amount of goods. Again, at the
present time at least, all questions of means have not
been brought within the range of science ; the estimation
of demand and the foreseeing of supply are matters not
yet reduced to any scientific basis. Besides, no man is
expert in all sciences, and the fact that he is familiar
with one may tend to hide from him his ignorance of
another. All this may be readily granted ; but all
it proves is that something besides scientific knowledge
is required for the competent conduct of affairs. Because
the man of science needs the help of the man trained in
commerce or administration, it does not follow that the
latter does not need the help of the former.

The attack on the practical value of science that we

170 WHAT IS SCIENCE?

are considering is best met by a counter-offensive. It
sounds plausible to maintain that those who have had
the greatest experience of any matter must know most
about it. But, like many other plausible doctrines, this
one is absolutely false. No popular saying is more
misleading than that we learn from experience ; really
the capacity of learning from experience is one of the
rarest gifts of genius, attained by humble folk only by
long and arduous training. Anyone who examines
carefully any subject concerning which popular beliefs
are prevalent, will always discover that those beliefs
are almost uniformly contradicted by the commonest
everyday experience. We shall not waste our time if
we devote a few pages to discovering the sources of
popular fallacies, and considering in what manner they
can be corrected by scientific investigation. When
we have established how little worthy of confidence is
" practical knowledge " we shall be in a position to see
the value of " theory."

POPULAR FALLACIES

The most frequent source of such fallacies is a disposi-
tion to accept without inquiry statements made by
other people. Error from this source is not wholly
avoidable ; except in the very few matters in which we
can interest ourselves, we must, if we are to avoid blank
ignorance, simply believe what we are told by the best
authority we can secure. And since nobody is always
right, we shall always believe some false doctrines
however carefully we choose our authority. But it
is very remarkable how people will go on believing things
on authority, when the weight of that authority is quite
unknown, and when their belief is flatly contradicted
by experience. I know a family, not without intelligence,
who, until their statement was challenged in a heated

THE APPLICATIONS OF SCIENCE 171

argument, always believed, on the authority of some
family tradition of unknown origin, that the walk they
took every Sunday afternoon was eight miles long ; and
yet a party which was not specially athletic, starting after
three, always accomplished it before five. A glance
at a map which was hanging in their house would have
shown them it was barely six miles. This will doubtless
seem a very extreme example concerning a very trivial
matter, but parallels can be found readily in matters
of considerable importance. During the war it was
almost a sufficient reason for the army chiefs to adopt
some device that it was known (or more often believed)
that the enemy made use of it ; and anyone who comes
into contact with unscientific managers of industry will
be amazed to find how largely their practice is based on
hearsay information, and how little evidence they have
that the information was reliable or even given in good
faith.

In matters which lie outside their own sphere, men of
science are often as credulous as anyone else ; but in
that sphere, if they are really men of science, intimately
acquainted with their study by the actual practice of
it, they cannot fail to have learnt how dangerous it is
to believe any statement, however firmly asserted by
high authority, unless they have tested it for themselves.
The necessity for the obtaining of information by direct
experiment is embedded in their nature, and no informa-
tion attained by other means will satisfy them per-
manently. The determination to believe what is true,
and not what other people assert to be true, is the first
and not the least important correction applied by science
to popular errors.

But if there were no other source of error, reliance on
hearsay would not be so dangerous, for our informants
would not be so likely to be wrong. There would still
be the possibility that they intended to mislead us,

172 WHAT IS SCIENCE?

though we may neglect that possibility for our purpose.
A more serious possibility would remain, namely, that
we had misinterpreted their information, and this is
actually the greatest danger in knowledge acquired at
second-hand. Thus the error about the length of the walk
quoted just now, doubtless arose from the fact that the
original Sunday walk was eight miles, and that a weaker
generation had abbreviated it to six. However, there
are other sources of error ; people do draw false conclu-
sions directly from experience ; and even Tf~we~~o5uTd
be sure that we had rightly understood an honest
informant, there would still be a danger that his informa-
tion was wrong. In discussing these other sources
and giving examples of them, it will be impossible to
distinguish them wholly from the first, for all popular
beliefs (from which many technical and professional
beliefs do not differ essentially) derive much of their
weight from their general prevalence. We can only
ask what fallacies predisposed men to these beliefs,
and thus enabled the beliefs to become prevalent.

The most prolific of these fallacies are false theories.
In discussing scientific theories in Chapter V, we saw
that in their ultimate nature they are not very different
from any other and unscientific attempt to explain things.
A theory which suggests that A and B might be connected
predisposes to the belief that A and B are connected. An
extreme example of unscientific theories is provided by
the superstitions and magical beliefs of primitive civiliza-
tion. They have ceased to be held explicitly, but they
still exert an influence, which is not generally appreciated,
on popular beliefs. Thus many people believe, and are
very indignant when the assertion is denied, that the
poker laid across the bars of the grate will draw up
the fire. The belief is based on the old doctrine of the
magical power of the cross, formed in this instance by
the poker and the bars ; old people can still be found

THE APPLICATIONS OF SCIENCE 173

who will say that the poker " keeps the witch up the
chimney/' Experiment would show that the poker has
no effect whatsoever ; but it is not easy to undertake
seriously, because the circumstances of " drawing up "
a fire are so indeterminate. Most popular weather-lore
has a similar origin in false theory ; people are ready to
think that the weather will change when the moon
changes, only because they think that the moon might
have some effect on the weather. Again, the persistent
feeling that there is some intimate connexion between
names, and the things of which they are names, leads to
curious credulity. " Rain before seven, fine before
eleven/' would never have become a popular saying had
it not been for the purely verbal jingle.1

It is scarcely too much to say that every popular
belief concerning such matters is false, and can be refuted
by experience which is directly accessible to those who
assert it ; and that the reason why such beliefs have
gained a hold can always be traced to some false theory
which, nowadays, only needs to be expressed to be
rejected. Their prevalence indicates clearly how little
the majority of mankind can be trusted to analyse their
experience carefully, and to base conclusions on that
experience and on nothing else. And it has been
admitted that the most careful and accurate science
does not base its conclusions only on experience ; in
science, too, analysis is guided by theory. But there is
this vast difference : though science may in the first
instance analyse experience and put forward laws guided
by theory and not by simple examination of facts, when
the analysis is completed and the laws suggested, it does
return and compare them with the facts. It is by this

1 There may be this truth in the saying that, in many parts of
England, continuous rain for four hours is unusual at any time of the
day or night. But if it is meant, as it usually is meant, that rain at
seven is more likely to be followed by fine weather than rain at six
or eight, then the saying is certainly false.

174 WHAT IS SCIENCE?

procedure that it has been able to establish true theories
which may be trusted, provisionally at least, in new
analysis. The practical man is apt to sneer at the
theorist ; but an examination of any of his most firmly-
rooted prejudices would show at once that he himself
is as much a theorist as the purest and most academic
student ; theory is a necessary instrument of thought
in disentangling the amazingly complex relations of the
external world. But while his theories are false because
he never tests them properly, the theories of science are
continually under constant test and only survive if they
are true. It is the practical man and not the student
of pure science who is guilty of relying on extrava-
gant speculation, unchecked by comparison with solid
fact.

^ Closely connected with the errors of false theories are
those which arise from false, or more often incomplete,
laws. Such laws are, of course, in themselves errors,
but they often breed errors much more serious. For
we have seen that the things between which laws assert
relations are themselves interconnected by laws ; if we
start with false laws we are sure to interpret our
experience on the wrong lines, because the things between
which we shall try to find laws will be such that no laws
can involve them. An example which we have used
before will make this source of error clear. The word
" steel " is used in all but the strictest scientific circles
to denote many different things ; or, in other words,
there is no law asserting the association of all the pro-
perties of all the things which are conventionally called
steel. Accordingly there can be no law, strictly true,
asserting anything about steel ; for though a law may be
found which is true of many kinds of steel, some kind
of " steel " can almost certainly be found of which it
is not true. If we want to find true laws involving the
materials all of which are conventionally termed steel,

THE APPLICATIONS OF SCIENCE 175

we must first differentiate the various kinds of steel
and seek laws which involve each kind separately.

Neglect of this precaution is one of the most frequent
causes of a failure to detect and to cure troubles encoun-
tered in industrial processes. The unscientific manager
regards as identical everything that is sold to him as steel ;
he regards as " water " everything that comes out of
the water main ; and as " gas " everything that comes
out of the gas main. He does not realize that these
substances, though called by the same name, may have
very different properties ; and when his customary
process does not lead to the usual result, he will probably
waste a great deal of time and money on far-fetched
ideas before he realizes that he did not get the same
result because he did not start with the same materials.
He can expect his processes to be governed by laws and
to lead invariably to the same result only if all the
materials and operations involved in those laws and
employed in those processes are themselves invariable ;
that is to say, if their constituent properties and events
are themselves associated by invariable laws. This is
a very obvious conclusion when it is pointed out, but
there is no conclusion more difficult to impress finally
on the practical man ignorant of science. He is misled
by words. Words are very useful when they really
represent ideas, but are a most terrible danger when they
do not. A word represents ideas, in the sense important
for the practical applications of science, only when the
, things which it is used to denote are truly collections of
properties or events associated by laws ; for it is only
then that the word can properly occur in one of the laws
on which all those applications depend. Perhaps the
most important service which science can render to
practical affairs is to insist that laws can only be expected
to hold between things which are themselves the expres-
sion of laws..

176 WHAT IS SCIENCE?

The last of the main sources of popular error is
connected with a peculiar form of law which brevity has
forbidden us to discuss hitherto. We have spoken of
laws as asserting invariable associations. Now, a very
slight acquaintance with science will suggest that this
view is unduly narrow ; it may seem that some laws in
almost all sciences (and almost all laws in some sciences)
assert that one event is associated with another, not
invariably, but usually or nearly always. Thus, if
meteorology, the science of weather, has any laws at all,
they would seem to be of this type ; nobody pretends
that it is possible at present, or likely to be possible in the
near future, to predict the weather exactly, especially
a long time ahead ; the most we can hope for is to discover
rules which will enable us generally to predict rightly.
Another instance may be found in the study of heredity.
It is an undoubted fact that, whether in plants, animals,
or human beings, children of the same parents generally
resemble each other and their parents more than they
do others not closely related. But even the great
progress in our knowledge of the laws of inheritance
which has been made in recent years has not brought
us near to a position in which it can be predicted (except
in a few very simple cases) what exactly will be the
property of each child of known parents. We know
some rules, but they are not the exact and invariable
rules which we have hitherto regarded as constituting
laws.

The pure scientific view of such laws is very interesting.
Put briefly it is that in such cases we have a mingling
of two opposed agencies. There are laws concerned in
such events, laws as strict and as invariable as those
which are regarded as typical, but they are acting
as it were on events governed not by law but by chance.
The result of any law or set of laws depends (cf. p. 167)
not only on the laws but on the events to which they are

THE APPLICATIONS OF SCIENCE 177

applied ; the irregularity that occurs in the study of the
weather or of heredity is an irregularity in such events.
Moreover, when science uses the conception of events
governed by chance, it means something much more
definite than is associated with that word by popular
use. We say in ordinary discourse that an event is due
to pure chance when we are completely ignorant whether
it will or will not happen. But complete ignorance
can never be a basis for knowledge, and the scientific
conception of chance, which does lead to knowledge,
implies only a certain limited degree of ignorance
associated with a limited degree of knowledge. It is
impossible here to discuss accurately what ignorance
and what knowledge are implied, but it may be said
roughly that the ignorance concerns each particular
happening of the event, while the knowledge concerns
a very long series of a great number of happenings. Thus,
scientifically, it is an even chance whether a penny falls
head or tail, because while we are perfectly ignorant
whether on each particular toss it will fall head or tail,
we are perfectly certain that in a sufficiently long series
of tosses, heads and tails will be nearly equally dis-
tributed.

When, therefore, science finds, as in the study of
weather or of heredity, phenomena which show some
regularity, but not complete regularity, it tries to analyse
them into perfectly regular laws acting on " chance "
material. And the first step in the analysis is always to
examine long series of the phenomena and to try to
discover in these long series regularities which are not
found in the individual members of them ; the regularity
will usually consist in each of the alternative phenomena
happening in a definite proportion of the trials. When
such regularity has been found, a second step can some-
times be taken and an analysis made into strict laws
applied to events which are governed by the particular

12

178 WHAT IS SCIENCE?

form of regularity which science regards as pure chance.
If that step can be taken, the scientific problem is solved,
for pure chance, like strict law, is one of the ultimate
conceptions of science. But there is often (as in
meteorology) a long interval between the first step and
the second, and in that interval all that is known is a
regularity in the long series which is usually called
a " statistical " law.

This procedure, like most scientific procedure, is
borrowed and developed from common sense ; but —
and this is the reason why it is mentioned here — it is
here that modern common sense lags most behind
scientific method. It is a familiar saying that statistics
can prove anything ; and so they can in the hands of
those not trained in scientific analysis. Statistical laws
are one of the most abundant sources of popular fallacies
which arise both from an ignorance of what such a law
means and a still greater ignorance of how it is to be
established. A statistical law does not state that some-
thing always happens, but that it happens more (or
less) frequently than something else ; the quotation of
instances of the thing happening are quite irrelevant
to a proof of the law, unless there is at the same time a
careful collection of instances in which it does not happen.
Moreover, the clear distinction between a true law and
a statistical law is not generally appreciated. A statis-
tical law, which is really scientific, is made utterly
fallacious in its application because it is interpreted as
if it predicted the result of individual trials.

For reasons which have been given already, many of
the laws that are most important in their practical
application are statistical laws ; and anyone, with a little
reflection, can suggest any number of examples of them.
Those which are generally familiar are usually entirely
fallacious (e.g. laws of weather and of heredity), and even
those which are true are habitually misapplied. The

THE APPLICATIONS OF SCIENCE 179

general misunderstanding of statistical laws, renders
them peculiarly liable, not only to their special fallacies,
but to all those which we have discussed before. False
theories and prejudices lead men to notice only those
instances which are favourable to the law they want to
establish ; they fail to see that, if it were a true law, a
single contrary instance would be sufficient to disprove
it, while, if it is a statistical law, favourable instances
prove nothing, unless contrary instances are examined
with equal care. They forget, too, that a statistical
law can never be the whole truth ; it may, for the time
being, represent all the truth we can attain ; but our efforts
should never cease, until the full analysis has been
effected, and the domain of strict law carefully separated
from that of pure chance. The invention of that method
of analysis, which leads to a possibility of prediction and
control utterly impossible while knowledge is still in the
statistical stage, is one of the things which makes science
indispensable to the conduct of the affairs of practical
life.

CONCLUSION

Our examination of the errors into which uninstructed
'reasoning is -liable to 'mislead mankind, affected by so
many prejudices and superstitions, shows immediately
how and why science is indispensable, if any valuable
lessons are j to , be drawn from the most ordinary
experience, jln the first place, science brings to the
analysis of such experience the .conception of definite,
positive, and fixed law. For the vague conception of a
law as a predominating, though variable, association,
always liable to be distorted by circumstances, or even,
like the laws of mankind, superseded by the vagaries of
some higher authority, science substitutes, as its basic
and most important conception, the association which
13

180 WHAT IS SCIENCE?

is absolutely invariable, unchanging, universal. We
may not always be successful in finding such laws, but
our firm belief that they are to be found never wavers ;
we never have the smallest reason to abandon our funda-
mental conviction that all events and changes, except
in so far as they are the direct outcome of volition (and
therefore immediately controllable), can be analysed into,
and interpreted by, laws of the strictest form. It is only
those who are guided by such a conviction that can hope
to bring order and form into the infinite complexity of
everyday experience.

However, such a conviction by itself would probably
be of little avail. If we only knew from the outset that
the analysis and explanation of experience could not be
effected, except by disentangling from it the strict laws
of science, every fresh problem would probably have to
await solution until it came to the notice of some great
genius ; for, as we noticed before, the discovery of a
wholly new law is one of the greatest achievements of
mankind. But we know much more ; the long series of
laws which have been discovered, indicate where new
laws are to be sought. We know that the terms involved
in a new law must themselves be associated by a law.
Moreover, the laws which define terms involved in other
laws, though numerous, form a well-recognized class ;
there are the laws defining various kinds of substances
or various species of living beings, those defining forces,
volumes, electric currents, the many forces of energy,
and all the various measurable quantities of physics ;
a complete list of them would fill a text-book, and yet
their number is finite and comprehended by all serious
students of the branches of science in which they are
involved. Such students know that, when they try to
analyse and explain new experience, it is between a
definite class of terms that the necessary laws must be
sought ; and that knowledge reduces the problem to

THE APPLICATIONS OF SCIENCE 181

one within the compass of a normal intellect, provided
it is well trained. He who seeks to solve the problem
without that knowledge and without the training on
which it is based, cannot hope for even partial success,
unless he can boast the powers of a Galileo or a Faraday.

And science provides yet another clue. It has estab-
lished theories as well as laws. Its theories do not cover
the full extent of its laws, and in some sciences little
guidance except that of " empirical " laws is available.
But where theories exist, they serve very closely to limit
the laws by means which it is worth while to try to
analyse experience ; no law contradictory of a firmly-
rooted theory is worth examining till all other alterna-
tives have been exhausted. Here uninstructed inquirers
are at a still greater disadvantage, compared with those
familiar with the results of science, for while many of the
terms involved in scientific laws are vaguely familiar
to every one, it is only those who have studied seriously,
who have any knowledge of theories.

Here a word of warning should be given. " Theory "
is always used in the book to mean the special class of
propositions discussed in Chapter V. When in popular
parlance " theory " is contrasted with " practice " it
is often not this kind of theory that is meant at all. The
plain man — I do not think this is an overstatement —
calls a " theory " anything he does not understand,
especially if the conclusions it is used to support are
distasteful to him. Arguments about matters in which
science is concerned, though they are denounced as wildly
" theoretical," often depend on nothing but firmly-
established law. It is only because he does not under-
stand " theory " that the plain man is apt to compare
it unfavourably with " practice," by which he means
what he can understand. The idea that something can
be " true in theory but false in practice " is due to
mere ignorance ; if any portion of " practice," about

182 WHAT IS SCIENCE?

which there is no doubt, is inconsistent with some
" theory," then the " theory " (whether it is a law or
what we call a theory) is false — and there is an end of it.
But it may easily happen, and does often happen, that
a " theory " is misinterpreted by those who fail to
understand it ; it appears to predict something incon-
sistent with practice only because its real meaning is
not grasped. It is certainly true that those who do not
understand " theory " had better leave it alone ; reliance
on misunderstood theory is certainly quite as dangerous
as reliance on uninstructed " practice."

And here we come to the conclusion about the relation
of science to everyday life, which it seems to me most
important to enforce. Those unversed in the ways of
science often regard it as a body of fixed knowledge con-
tained in text -books and treatises, from which anyone
who takes the trouble can extract all the information on
any subject which science has to offer ; they think of it
as something that can be learnt as the multiplication
table can be learnt, and consider that anyone who has
" done " science at school or college is complete master
of its mysteries. Nothing could be further from the
truth regarding science applied to practical problems.
It is scarcely ever possible, even for the most learned
student, to offer a complete and satisfying explanation
of any difficulty, merely on the basis of established know-
ledge ; there is also some element in the problem which
has not yet engaged scientific attention. Applied science,
like pure science, is not a set of immutable principles
and propositions ; it is rather an instrument of thought
and a way of thinking. Every practical problem is really
aT problem irTresearch, leading to the advancement of
pure learning as well as to material efficiency ; indeed
almost all the problems by the solution of which science
has actually advanced have been suggested, more or less
directly, by the familiar experiences of everyday life

THE APPLICATIONS OF SCIENCE 183

This tremendous instrument of research can be mastered,
this new way of thinking can be acquired, only by long
training and laborious exercise. It is not, or it ought not
to be, the academic student in the pure refined air of the
laboratory who makes the knowledge, and the hard-
handed and hard-headed worker who applies it to its
needs. The man who can make new knowledge is the
man, and the only man, to apply it.

Pure and applied science are the roots and the branches
of the tree of ex^ejimentRl knowledge ; theory and
practice are inseparably interwoven, and cannot be torn
asunder without grave injury to both. The intellectual
and the material health of society depend on the main-
tenance of their close connexion. A few years ago there
was a tendency for true science to be confined to the
laboratory, for its students to become thin-blooded,
deprived of the invigorating air of industrial life, while
industry wilted from neglect. To-day there are perhaps
some signs of an extravagant reaction ; industrial
science receives all the support and all the attention, while
the universities, the nursing mothers of all science and
all learning, are left to starve. The danger of rushing
from one extreme to another will not be avoided until
there is a general consciousness of what science means,

pf infftllprfnal gatlSfaCtfon Uld .._SSL a

_

means of attaining material desires. We cannot all be —
ft is not desirable that weTall should be — close students
of science ; but we can all appreciate in some measure
what are its aims, its methods, its uses. Science, like
art, should not be something extraneous, added as a
decoration to the other activities of existence ; it should
be part of them, inspiring our most trivial actions as well
as our noblest thoughts.

INDEX

AGREEMENT, Universal, 27-36,

38, 48, 86, 92, 107, 108
Ampere, 75, 120, 155
Anthropology, 12
Archimedes, 120, 125
Association, Invariable, 41, 42,

44, 49, 55, 62-66, 68, 105, 175,

176

BACH, 73
Bacon, 29, 72
Beethoven, 73, 102, 103
Bohr, 156
Botany, 13, 56, 75
Boyle, 75, 81, 82, 87

CALCULATION, 144-153

Cause and effect, 39, 40, 49-55

Cavendish, 120

Certainty, 59, 60

Chance, 176-179

Chemistry, 13, 75

Colour, 33, 34, 128

Copernicus, 5, 94, 99

Coulomb, 1 20 ,

Counting, 112-116

DENSITY, 125-133, 144-146
Dreams, 22, 23
Dynamics, 84-86, 99, 100

ECONOMICS, 15
Einstein, 102, 156, 157
Evolution, Theory of, 5, 96, 97
Experiment, 54
Explanation, 77-80
External world, 17-27, 35, 38

FALLACIES, Popular, 170-179
Faraday, 155, 160, 181
" Function," 142
Future, 59, 60, 64, 65, 70

GALILEO, 73, 75, 93, 94, 99, 100,

120, 181

Gases, Properties of, 81, 82, 86
Gay-Lussac, 81, 82, 87
Geology, 13, 96

Gravitation, 99-103
Greek learning, 6-8, 120

HALLUCINATIONS, 22, 23, 32
Hardness, 128, 133
Heredity, 176-178
Hertz, 156
History, 7, 15, 37, 61
Huxley, 10

INDUCTION, 76

KEPLER, 99, 100
Kirchhoff, 120

LAVOISIER, 73, 75

Laws, Adaptation of, 64-67

, Analysis of, 43-46

, Bases of science, 37-39

, Development of, 40-49

, Discovery of, 58-76

distinguished from

theories, 85, 86

, Explanation of, 77-108

, Form of, 74-76

of measurement, 119-122

, Numerical, 52, 75, 76, 129-

132, I35-M4

, Object of, 67-69

, Scientific, 37-57

suggested by theory, 88, 91

, Value of, 69-71, 166-168

Linnaeus, 75
Lucretius, 93

MAGNETS, 41

Marconi, 156

Mathematics, 7, 109, 142-157

Matter, 13

Maxwell, 155, 156

Means and ends, 160-163

Measurement, 109-134

, derived, 124-132, 142-144

, Laws of, 119-122

of length, 118

of number, m-ii6

, value of, 132-134

of weight, 117

185

186

WHAT IS SCIENCE?

Medicine, 76
Meteorology, 76
Mill, 72
Mineralogy, 56
Molecules, 81, 82, 87, 88, 104
Multiplication, 122, 123, 136-
141

NAPOLEON, 41, 42
Nature, 10-12, 15-27

, Uniformity of, 61, 62

Newton, 73, 93, 94, 99-103, 148-

152

Number, m-ii6, 137, 138
Numerals, 112, 115, 116, 126,

137, 138
, Order of, 126, 129

OERSTED, 120

Ohm, 52, 53, 120

" Other people," 23-27

PASTEUR, 160
Pendulum, 152
Philosophy, 7, 9-11, 28

, Natural, 9-11

Physics, 13, 14, 75
Physiology, 13
Politics, 162, 163
Prediction, 59, 69-71, 87-89, 91
Psychology, 12

.QUALITY and quantity, 125

SCIENCE, Branches of, 13

, Certainty of, 164-170, 179

, Classificatory, 56, 57

and common sense, 17, 29,

48, 92, 93

Science and imagination, 73-76,

90-94, 97-104

, Limitations of, 160-164

and logic, 47, 83, 145-147

and other studies, 5-9

and practice, 158-183

pure and applied, 1-5, 158-

160
Scientific, 9

management, 163

Sensations, 16-19, 20-23, 24-27,

86

Sommerfeld, 156, 157
" Square," 140
Statistics, 176-179
Steel, 40, 41, 43-46, 52, 174
Substance, 45, 66, 74, 76, 93
Symbols, 154-156

THEORY, Analogies in, 94-97
distinguished from laws,

85, 86

of gases, 81, 82, 96, 97

, Invention of, 89-94

, Mathematical, 153-157

, Mechanical, 95

and " practice," 164, 170,

181, 182

, Reality of, 104-108

, Scientific, 81-85

, Unscientific, 92, 93, 172-

*74
, Value of, 86-89

WEATHER, 173, 175, 177, 178
Will, 20, 26

ZOOLOGY, 13, 56

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