OUR KNOWLEDGE OF THE EXTERNAL WORLD

OUR KNOWLEDGE OF THE
EXTERNAL WORLD

BY BERTRAND RUSSELL
AUTHORITY AND IBB INDIVIDUAL

HUMAN KNOWLEDGE: in SCOPE AND IIMITS

HISTORY OF WESTERN PHILOSOPHY

THE PRINCIPLES OP MATHEMATICS

INTRODUCTION TO MATHEMATICAL PHILOSOPHY

THE ANALYSIS OF MIND

AN OUTLINB OF PHILOSOPHY

THE PHILOSOPHY OF LEIBNIZ

AN INQUIRY INTO MEANING AND TRUTH

UNPOPULAR ESSAYS

IN PRAISE OF IDLENESS
THE CONQUEST OF HAPPINESS

MYSTICISM AND LOGIC

THE SCIENTIFIC OUTLOOK

MARRIAGE AND MORALS

EDUCATION AND THE SOCIAL ORDER

ON EDUCATION

FREEDOM AND ORGANIZATION, 1814-1914

PRINCIPLES OF SOCIAL RECONSTRUCTION

ROADS TO FREEDOM

JUSTICE IN WAR-TIME

FREE THOUGHT AND OFFICIAL PROPAGANDA

BERTRAND RUSSELL

OUR KNOWLEDGE

OF THE
EXTERNAL WORLD

AS A FIELD FOR SCIENTIFIC
METHOD IN PHILOSOPHY

GEORGE ALLEN & U N WIN LTD
RUSKIN HOUSE, 40 MUSEUM STREET, LONDON

8 SCIENTIFIC METHOD IN PHILOSOPHY

of mathematical physics. I have been made aware
of the importance of this problem by my friend and
collaborator Dr. Whitehead, to whom are due almost
all the differences between the views advocated here
and those suggested in The Problems of Philosophy*
I owe to hi the definition of points, the suggestion
for the treatment of instants and " things," and the
whole conception of the world of physics as a construc-
tion rather than an inference. What is said on these
topics here is, in fact, a rough preliminary account
of the more precise results which he is giving in the
fourth volume of our Principia Mathematical It will
be seen that if his way of dealing with these topics is
capable of being successfully carried through, a wholly
new light is thrown on the time-honoured controversies
of realists and idealists, and a method is obtained of
solving all that is soluble in their problem.

The speculations of the past as to the reality or
unreality of the world of physics were baffled, at the
outset, by the absence of any satisfactory theory of
the mathematical infinite. This difficulty has been
removed by the work of Georg Cantor. But the
positive and detailed solution of the problem by means
of mathematical constructions based upon sensible
objects as data has only been rendered possible by the
growth of mathematical logic, without which it is
practically impossible to manipulate ideas of the
requisite abstractness and complexity. This aspect,
which is somewhat obscured in a merely popular
outline such as is contained in the following lectures,
will become plain as soon as Dr. Whitehead's work is
published. In pure logic, which, however, will be very

LondonandNewYork, 1912 ("Home University Library"),
a The first volume was published at Cambridge in 1910, the
second in 1912, and the third in 1913.

PREFACE 9

briefly discussed in these lectures, I have had the
benefit of vitally important discoveries, not yet pub-
lished, by my friend Mr. Ludwig Wittgenstein.

Since my purpose was to illustrate method, I have
included much that is tentative and incomplete, for
it is not by the study of finished structures alone that
the manner of construction can be learnt. Except
in regard to such matters as Cantor's theory of infinity,
no finality is claimed for the theories suggested ; but
I believe that where they are found to require modi-
fication, this will be discovered by substantially the
same method as that which at present makes them
appear probable, and it is on this ground that I ask the
reader to be tolerant of their incompleteness.

CAMBRIDGE,
June 1914.

CONTENTS

LECTURE PAGE

I. Current Tendencies 13

II. Logic as the Essence of Philosophy 42

III. On Our Knowledge of the External World , 70

IV. The World of Physics and the World of Sense 106
V. The Theory of Continuity 135

VI. The Problem of Infinity Considered Historically 159

VII. The Positive Theory of Infinity 189

VIII. On the Notion of Cause, with Applications to

the Free-Will Problem 214

Index 247

OUR KNOWLEDGE OF THE
EXTERNAL WORLD

LECTURE I
CURRENT TENDENCIES

PHILOSOPHY, from the earliest times, has made
greater ml aim a, and achieved fewer results, than
any other branch, of learning. Ever since Thales
said that all is water, philosophers have been ready
with glib assertions about the sum-total of things ;
and equally glib denials have come from other philo-
sophers ever since Thales was contradicted by Anaxi-
mander. I believe that the time has now arrived
when this unsatisfactory state of things can be brought
to an end. In the following course of lectures I
shall try, chiefly by taking certain special problems as
examples, to indicate wherein the claims of philo-
sophers have been excessive, and why their achieve-
ments have not been greater. The problems and the
method of philosophy have, I believe, been miscon-
ceived by all schools, many of its traditional problems
being insoluble with our means of knowledge, while
other more neglected but not less important problems
can, by a more patient and more adequate method, be
solved with all the precision and certainty to which the
most advanced sciences have attained.

Among present-day philosophies, we may distin-

14 SCIENTIFIC METHOD IN PHILOSOPHY

guish three principal types, often combined in varying
proportions by a single philosopher, but in essence
and tendency distinct. The first of these, which I
shall call the classical tradition, descends in the main
from Kant and Hegel ; it represents the attempt to
adapt to present needs the methods and results of
the great constructive philosophers from Plato down-
wards. The second type, which may be called evolu-
tionism, derived its predominance from Darwin,
and must be reckoned as having had Herbert Spencer
for its first philosophical representative ; but in
recent times it has become, chiefly through William
James and M. Bergson, far bolder and far more search-
ing in its innovations than it was in the hands of
Herbert Spencer. The third type, which may be called
" logical atomism " for want of a better name, has
gradually crept into philosophy through the critical
scrutiny of mathematics. This type of philosophy,
which is the one that I wish to advocate, has not as
yet many whole-hearted adherents, but the "new
realism" which owes its inception to Harvard is
very largely impregnated with its spirit. It repre-
sents, I believe, the same kind of advance as was
introduced into physics by Galileo : the substitution of
piecemeal, detailed, and verifiable results for large
untested generalities recommended only by a certain
appeal to imagination. But before we can understand
the changes advocated by this new philosophy, we
must briefly examine and criticize the other two types
with which it has to contend.

A. THE CLASSICAL TRADITION

Twenty years ago, the classical tradition, having
vanquished the opposing tradition of the English

CURRENT TENDENCIES 15

empiricists, held almost unquestioned sway in all
Anglo-Saxon universities. At the present day, though
it is losing ground, many of the most prominent
teachers still adhere to it. In academic France, in
spite of M. Bergson, it is far stronger than all its
opponents combined ; and in Germany it had many
vigorous advocates. Nevertheless, it represents on
the whole a decaying force, and it has failed to adapt
itself to the temper of the age. Its advocates are,
in the main, those whose extra-philosophical know-
ledge is liteniry, rather than those who have felt the
inspiration of science. There are, apart from reasoned
arguments, certain general intellectual forces against
it the same general forces which are breaking down
the other great syntheses of the past, and making our
age one of bewildered grouping where our ancestors
walked in the clear daylight of unquestioning certainty.
The original impulse out of which the classical
tradition developed was the naive faith of the Greek
philosophers in the omnipotence of reasoning. The
discovery of geometry had intoxicated them, and its
a priori deductive method appeared capable of universal
application. They would prove, for instance, that
all reality is one, that there is no such thing as change,
that the world of sense is a world of mere illusion ;
and the strangeness of their results gave them no
qualms because they believed in the correctness of
their reasoning. Thus it came to be thought that
by mere thinking the most surprising and important
truths concerning the whole of reality could be estab-
lished with a certainty which no contrary observations
could shake. As the vital impulse of the early philo-
sophers died away, its place was taken by authority
and tradition, reinforced, in the Middle Ages and
almost to our own day, by systematic theology.

16 SCIENTIFIC METHOD IN PHILOSOPHY

Modern philosophy, from Descartes onwards, though
not bound by authority like that of the Middle Ages,
still accepted more or less uncritically the Aristotelian
logic. Moreover, it still believed, except in Great
Britain, that a priori reasoning could reveal otherwise
undiscoverable secrets about the universe, and could
prove reality to be quite different from what, to direct
observation, it appears to be. It is this belief, rather
than any particular tenets resulting from it, that I
regard as the distinguishing characteristic of the
classical tradition, and as hitherto the main obstacle
to a scientific attitude in philosophy.

The nature of the philosophy embodied in the
classical tradition may be made clearer by taking a
particular exponent as an illustration. For this
purpose, let us consider for a moment the doctrines
of Mr. Bradley, who is probably the most distinguished
British representative of this school. Mr. Bradley's
Appearance and Reality is a book consisting of two
parts, the first called Appearance, the second Reality.
The first part examines and condemns almost all that
makes up our everyday world : things and qualities,
relations, space and time, change, causation, activity,
the self. All these, though in some sense facts which
qualify reality, are not real as they appear. What
is real is one single, indivisible, timeless whole, called
the Absolute, which is in some sense spiritual, but does
not consist of souls, or of thought and will as we know
them. And all this is established by abstract logical
reasoning professing to find self-contradictions in the
categories condemned as mere appearance, and to
leave no tenable alternative to the kind of Absolute
which is finally affirmed to be real.

One brief example may suffice to illustrate Mr. Brad-
ley's method. The world appears to be full of

CURRENT TENDENCIES 17

many things with various relations to each other
right and left, before and after, father and son, and
so on. But relations, according to Mr. Bradley, axe
found on examination to be self-contradictory and
therefore impossible. He first argues that, if there
are relations, there must be qualities between which
they hold. This part of his argument need not detain
us. He then proceeds :

"But how the relation can stand to the qualities
is, on the other side, unintelligible. If it is nothing
to the qualities, then they are not related at all;
and, if so, as we saw, they have ceased to be qualities,
and their relation is a nonentity. But if it is to be
something to them, then clearly we shall require a
new connecting relation. For the relation hardly
can be the mere adjective of one or both of its terms ;
or, at least, as such it seems indefensible. And,
being something itself, if it does not itself bear a rela-
tion to the terms, in what intelligible way will it
succeed in being anything to them ? But here again
we are hurried off into the eddy of a hopeless process,
since we are forced to go on finding new relations
without end. The links are united by a link, and this
bond of union is a link which also has two ends ; and
these require each a fresh IfriTr to connect them with
the old. The problem is to find how the relation
can stand to its qualities, and this problem is
insoluble." I

I do not propose to examine this argument in

detail, or to show the exact points where, in my opinion,

it is fallacious. I have quoted it only as an example

of method. Most people will admit, I think, that it

is calculated to produce bewilderment rather than

conviction, because there is more likelihood of error

1 Appearance and Reality, pp. 32-33.

2

i8 SCIENTIFIC METHOD IN PHILOSOPHY

in a very subtle, abstract, and difficult argument
than in so patent a fact as the interrelatedness of the
things in the world. To the early Greeks, to whom
geometry was practically the only known science, it.
was possible to follow reasoning with assent even
when it led to the strangest conclusions. But to us,
with our methods of experiment and observation, our
knowledge of the long history of a priori errors refuted
by empirical science, it has become natural to suspect
a fallacy in any deduction of which the conclusion
appears to contradict patent facts. It is easy to
carry such suspicion too far, and it is very desirable, if
possible, actually to discover the exact nature of the
error when it exists. But there is no doubt that what
we may call the empirical outlook has become part
of most educated people's habit of mind ; and it is
this, rather than any definite argument, that has
diminished the hold of the classical tradition upon
students of philosophy and the instructed public
generally.

The function of logic in philosophy, as I shall try
to show at a later stage, is all-important ; but I do
not think its function is that which it has in the classi-
cal tradition. In that tradition, logic becomes con-
structive through negation. Where a number of
alternatives seem, at first sight, to be equally possible,
logic is made to condemn all of them except one,
and that one is then pronounced to be realized in the
actual world. Thus the world is constructed by means
of logic, with little or no appeal to concrete experience.
The true function of logic is, in my opinion, exactly
the opposite of this. As applied to matters of experi-
ence, it is analytic rather than constructive ; taken
a priori, it shows the possibility of hitherto unsus-
pected alternatives more often than the impossibility

CURRENT TENDENCIES 19

of alternatives which seemed prima fade possible.
Thus, while it liberates imagination as to what the
world may be, it refuses to legislate as to what the
world is. This change, which has been brought about
by an internal revolution in logic, has swept away the
ambitious constructions of traditional metaphysics,
even for those whose faith in logic is greatest ; while
to the many who regard logic as a chimera the para-
doxical systems to which it has given rise do not seem
worthy even of refutation. Thus on all sides these
systems have ceased to attract, and even the philo-
sophical world tends more and more to pass them by.

One or two of the favourite doctrines of the school
we are considering may be mentioned to illustrate
the nature of its claims. The universe, it tells us, is
an " organic unity," like an animal or a perfect work
of art. By this it means, roughly speaking, that all
the different parts fit together and co-operate, and
are what they are because of their place in the whole.
This belief is sometimes advanced dogmatically,
while at other times it is defended by certain logical
arguments, If it is true, every part of the universe
is a microcosm, a miniature reflection of the whole.
If we knew ourselves thoroughly, according to this
doctrine, we should know everything. Common sense
would naturally object that there are people say
in China with whom our relations are so indirect
and trivial that we cannot infer anything important
as to them from any fact about ourselves. If there
are living beings in Mars or in more distant parts of
the universe, the same argument becomes even stronger.
But further, perhaps the whole contents of the space
and time in which we live form only one of many
universes, each seeming to itself complete. And thus
the conception of the necessary unity of all that is

20 SCIENTIFIC METHOD IN PHILOSOPHY

resolves itself into the poverty of imagination, and a
freer logic emancipates us from the strait-waistcoated
benevolent institution which idealism palms off as the
totality of being.

Another very important doctrine held by most,
though not all, of the school we are examining is the
doctrine that all reality is what is called " mental "
or "spiritual," or that, at any rate, all reality is
dependent for its existence upon what is mental.
This view is often particularized into the form which
states that the relation of knower and known is funda-
mental, and that nothing can exist unless it either
knows or is known. Here again the same legislative
function is ascribed to a priori argumentation : it
is thought that there are contradictions in an unknown
reality. Again, if I am not mistaken, the argument
is fallacious, and a better logic will show that no limits
can be set to the extent and nature of the unknown.
And when I speak of the unknown, I do not mean
merely what we personally do not know, but what
is not known to any mind. Here as elsewhere, while
the older logic shut out possibilities and imprisoned
imagination within the walls of the familiar, the newer
logic shows rather what may happen, and refuses to
decide as to what must happen.

The classical tradition in philosophy is the last sur-
viving child of two very diverse parents : the Greek
belief in reason, and the mediaeval belief in the tidi-
ness of the universe. To the schoolmen, who lived
amid wars, massacres, and pestilences, nothing
appeared so delightful as safety and order. In their
idealising dreams, it was safety and order that they
sought : the universe of Thomas Aquinas or Dante is
as small and neat as a Dutch interior. To us, to
whom safety has become monotony, to whom the

CURRENT TENDENCIES 21

primeval savageries of nature are so remote as to
become a mere pleasing condiment to our ordered
routine, the world of dreams is very different from what
it was amid the wars of Guelf and Ghibelline. Hence
William James's protest against what he calls the
" block universe " of the classical tradition ; hence
Nietzsche's worship of force ; hence the verbal blood- '
thirstiness of many quiet literary men. The barbaric
substratum of human nature, unsatisfied in action,
finds an outlet in imagination. In philosophy, as
elsewhere, this tendency is visible ; and it is this, rather
than formal argument, that has thrust aside the classical
tradition for a philosophy which fancies itself more
virile and more vital. 1

B. EVOLUTIONISM

Evolutionism, in one form or another, is the pre-
vailing creed of our time. It dominates our politics,
our literature, and not least our philosophy. Nietzsche,
pragmatism, Bergson, are phases in its philosophic
development, and their popularity far beyond the
circles of professional philosophers shows its conso-
nance with the spirit of the age. It believes itself
firmly based on science, a liberator of hopes, an iospirer
of an invigorating faith in human power, a sure anti-
dote to the ratiocinative authority of the Greeks and
the dogmatic authority of mediaeval systems. Against
so fashionable and so agreeable a creed it may seem
useless to raise a protest ; and with much of its spirit
every modern man must be in sympathy. But I
think that, in the intoxication of a quick success, much
that is important, and vital to a true understanding of
the universe has been forgotten. Something of Hellenism
1 Written before August 1914.

22 SCIENTIFIC METHOD IN PHILOSOPHY

must be combined with the new spirit before it can
emerge from the ardour of youth into the wisdom of
manhood. And it is time to remember that biology
is neither the only science, nor yet the model to which
all other sciences must adapt themselves. Evolu-
tionism, as I shall try to show, is not a truly scientific
philosophy, either in its method or in the problems
which it considers. The true scientific philosophy
is something more arduous and more aloof, appealing
to less mundane hopes, and requiring a severer dis-
cipline for its successful practice.

Darwin's Origin of Species persuaded the world
that the difference between different species of animals
and plants is not the fixed, immutable difference
that it appears to be. The doctrine of natural kinds,
which had rendered classification easy and definite,
which was enshrined in the Aristotelian tradition,
and protected by its supposed necessity for orthodox
dogma, was suddenly swept away for ever out of the
biological world. The difference between man and
the lower animals, which to our human conceit appears
enormous, was shown to be a gradual achievement,
involving intermediate beings who could not with
certainty be placed either within or without the human
family. The sun and planets had already been shown
by Laplace to be very probably derived from a primi-
tive more or less undifferentiated nebula. Thus the
old fixed landmarks became wavering and indistinct,
and all sharp outlines were blurred. Things and
species lost their boundaries, and none could say where
they began or where they ended.

But if human conceit was staggered for a moment
by its kinship with the ape, it soon found a way to
reassert itself, and that way is the " philosophy " of
evolution. A process which led from the amoeba to

CURRENT TENDENCIES 23

man appeared to the philosophers to be obviously a
progress though whether the amoeba would agree
with this opinion is not known. Hence the cycle of
changes which science had shown to be the probable
history of the past was welcomed as revealing a law
of development towards good in the universe an
evolution or unfolding of an ideal slowly embodying
itself in the actual. But such a view, though it might
satisfy Spencer and those whom we may call Hegelian
evolutionists, could not be accepted as adequate by
the more whole-hearted votaries of change. An ideal
to which the world continuously approaches is, to
these minds, too dead and static to be inspiring. Not
only the aspirations, but the ideal too, must change
and develop with the course of evolution ; there must
be no fixed goal, but a continual fashioning of fresh
needs by the impulse which is life and which alone
gives unity to the process.

Ever since the seventeenth century, those whom
William James described as the " tender-minded "
have been engaged in a desperate struggle with the
mechanical view of the course of nature which physical
science seems to impose. A great part of the attractive-
ness of the classical tradition was due to the partial
escape from mechanism which it provided. But now,
with the influence of biology, the " tender-minded "
believe that a more radical escape is possible, sweeping
aside not merely the laws of physics, but the whole
apparently immutable apparatus of logic, with its
fixed concepts, its general principles, and its reasonings
which seem able to compel even the most unwilling
assent. The older kind of teleology, therefore, which
regarded the End as a fixed goal, already partially
visible, towards which we were gradually approaching,
is rejected by M. Bergson as not allowing enough for

24 SCIENTIFIC METHOD IN PHILOSOPHY

the absolute dominion of change. After explaining
why he does not accept mechanism, he proceeds : t

" But radical finalism is quite as unacceptable, and
for the same reason. The doctrine of teleology, in
its extreme form, as we find it in Leibniz for example,
implies that things and beings merely realize a pro-
gramme previously arranged. But if there is nothing
unforeseen, no invention or creation in the universe,
time is useless again. As in the mechanistic hypo-
thesis, here again it is supposed that all is given.
Finalism thus understood is only inverted mechanism.
It springs from the same postulate, with this sole
difference, that in the movement of our finite intellects
along successive things, whose successiveness is reduced
to a mere appearance, it holds in front of us the light
with which it claims to guide us, instead of putting it
behind. It substitutes the attraction of the future
for the impulsion of the past. But succession remains
none the less a mere appearance, as indeed does
movement itself. In the doctrine of Leibniz, time is
reduced to a confused perception, relative to the human
standpoint, a perception which would vanish, like a
rising mist, for a mind seated at the centre of things.

" Yet finalism is not, like mechanism, a doctrine
with fixed rigid outlines. It admits of as many inflec-
tions as we like. The mechanistic philosophy is to
be taken or left : it must be left if the least grain of
dust, by straying from the path foreseen by mechanics,
should show the slightest trace of spontaneity. The
doctrine of final causes, on the contrary, will never
be definitively refuted. If one form of it be put aside,
it will take another. Its principle, which is essentially
psychological, is very flexible. It is so extensible,
and thereby so comprehensive, that one accepts some-
* Creative Evolution, Tfrigijgh translation, p. 41.

CURRENT TENDENCIES 25

thing of it as soon as one rejects pure mechanism.
The theory we shall put forward in this book will
therefore necessarily partake of finalism to a certain
extent."

M. Bergson's form of finalism depends upon his
conception of life. Life, in his philosophy, is a con-
tinuous stream, in which all divisions are artificial
and unreal. Separate things, beginnings and endings,
are mere convenient fictions : there is only smooth,
unbroken transition. The beliefs of to-day may count
as true to-day, if they cany us along the stream ; but
to-morrow they will be false, and must be replaced
by new beliefs to meet the new situation. All our
thinking consists of convenient fictions, imaginary
congealings of the stream : reality flows on in spite
of all our fictions, and though it can be lived, it cannot
be conceived in thought. Somehow, without explicit
statement, the assurance is slipped in that the future,
though we cannot foresee it, will be better than the
past or the present : the reader is like the child who
expects a sweet because it has been told to open its
mouth and shut its eyes. Logic, mathematics, physics,
disappear in this philosophy, because they are too
" static " ; what is real is an impulse and movement
towards a goal which, like the rainbow, recedes as we
advance, and makes every place different when we
reach it from what it appeared to be at a distance.

Now I do not propose at present to enter upon a
technical examination of this philosophy. At present
I wish to make only two criticisms of it first, that
its truth does not follow from what science has ren-
dered probable concerning the facts of evolution, and
secondly, that the motives and interests which inspire
it are so exclusively practical, and the problems with
which it deals are so special, that it can hardly be

26 SCIENTIFIC METHOD IN PHILOSOPHY

regarded as really touching any of the questions that
to my mind constitute genuine philosophy.

(i) What biology has rendered probable is that the
diverse species arose by adaptation from a less differen-
tiated ancestry. This fact is in itself exceedingly
interesting, but it is not the kind of fact from which
philosophical consequences follow. Philosophy is
general, and takes an impartial interest in all that
exists. The changes suffered by minute portions of
matter on the earth's surface are very important to us
as active sentient beings ; but to us as philosophers
they have no greater interest than other changes in
portions of matter elsewhere. And if the changes on
the earth's surface during the last few minions of years
appear to our present ethical notions to be in the
nature of a progress, that gives no ground for believing
that progress is a general law of the universe. Except
under the influence of desire, no one would admit for
a moment so crude a generalization from such a tiny
selection of facts. What does result, not specially
from biology, but from all the sciences which deal with
what exists, is that we cannot understand the world
unless we can understand change and continuity.
This is even more evident in physics than it is in
biology. But the analysis of change and continuity is
not a problem upon which either physics or biology
throws any light: it is a problem of a new kind,
belonging to a different kind of study. The question
whether evolutionism offers a true or a false answer to
this problem is not, therefore, a question to be solved
by appeals to particular facts, such as biology and
physics reveal. In assuming dogmatically a certain
answer to this question, evolutionism ceases to be
scientific, yet it is only in touching on this question
that evolutionism reaches the subject-matter of philo

CURRENT TENDENCIES 27

sophy. Evolutionism thus consists of two parts :
one not philosophical, but only a hasty generalization
of the kind which the special sciences might hereafter
confirm or confute; the other not scientific, but a
mere unsupported dogma, belonging to philosophy by
its subject-matter, but in no way deducible from the
facts upon which evolutionism relies.

(2) The predominant interest of evolutionism is in
the question of human destiny, or at least of the
destiny of life. It is more interested in morality
and happiness than in knowledge for its own sake.
It must be admitted that the same may be said of
many other philosophies, and that a desire for the
kind of knowledge which philosophy really can give
is very rare. But if philosophy is to become scientific
and it is our object to discover how this can be
achieved it is necessary first and foremost that philo-
sophers should acquire the disinterested intellectual
curiosity which characterizes the genuine man of
science. Knowledge concerning the future which is
the kind of knowledge that must be sought if we are
to know about human destiny is possible within
certain narrow limits. It is impossible to say how
much the limits may be 'enlarged with the progress
of science. But what is evident is that any proposi-
tion about the future belongs by its subject-matter
to some particular science, and is to be ascertained,
if at all, by the methods of that science. Philosophy
is not a short cut to the same kind of results as those
of the other sciences : if it is to be a genuine study,
it must have a province of its own, and aim at results
which the other sciences can neither prove nor disprove.

The consideration that philosophy, if there is such
a study, must consist of propositions which could
not occur in the other sciences, is one which has very

28 SCIENTIFIC METHOD IN PHILOSOPHY

far-reaching consequences. All the questions which
have what is called a human interest such, for
example, as the question of a future life belong, at
least in theory, to special sciences, and are capable,
at least in theory, of being decided by empirical
evidence. Philosophers have too often, in the past,
permitted themselves to pronounce on empirical
questions, and found themselves, as a result, in dis-
astrous conflict with well-attested facts. We must,
therefore, renounce the hope that philosophy can
promise satisfaction to our mundane desires. What
it can do, when it is purified from all practical taint,
is to help us to understand the general aspects of the
world and the logical analysis of familiar but complex
things. Through this achievement, by the suggestion
of fruitful hypotheses, it may be indirectly useful in
other sciences, notably mathematics, physics, and
psychology. But a genuinely scientific philosophy
cannot hope to appeal to any except those who have
the wish to understand, to escape from intellectual
bewilderment. It offers, in its own domain, the kind
of satisfaction which the other sciences offer. But
it does not offer, or attempt to offer, a solution of the
problem of human destiny, or of the destiny of the
universe.

Evolutionism, if what has been said is true, is to be
regarded as a hasty generalization from certain rather
special facts, accompanied by a dogmatic rejection
of all attempts at analysis, and inspired by interests
which are practical rather than theoretical. In spite,
therefore, of its appeal to detailed results in various
sciences, it cannot be regarded as any more*genuinely
scientific than the classical tradition which it has
replaced. How philosophy is to be rendered scientific,
and what is the true subject-matter of philosophy,

CURRENT TENDENCIES 29

I shall try to show first by examples of certain achieved
results, and then more generally. We will begin
with the problem of the physical conceptions of space
and time and matter, which, as we have seen, are
challenged by the contentions of the evolutionists.
That these conceptions stand in need of reconstruc-
tion will be admitted, and is indeed increasingly urged
by physicists themselves. It will also be admitted that
the reconstruction must take more account of change
and the universal flux than is done in the older
mechanics with its fundamental conception of an
indestructible matter. But I do not think the recon-
struction required is on Bergsonian lines, nor do I
think that his rejection of logic can be anything but
harmful. I shall not, however, adopt the method
of explicit controversy, but rather the method of
independent inquiry, starting from what, in a pre-
philosophic stage, appear to be facts, and keeping
always as dose to these initial data as the requirements
of consistency will permit.

Although explicit controversy is almost always
fruitless in philosophy, owing to the fact that no two
philosophers ever understand one another, yet it
seems necessary to say something at the outset in
justification of the scientific as against the mystical
attitude. Metaphysics, from the first, has been
developed by the union or the conflict of these two
attitudes. Among the earliest Greek philosophers,
the lonians were more scientific, and the Sicilians more
mystical. 1 But among the latter, Pythagoras, for
example, was in himself a curious mixture of the two
tendencies: the scientific attitude led him to his
proposition on right-angled triangles, while his mystic
insight showed him that it is wicked to eat beans.
* Cf. Burnet, Early Greek Philosophy, pp. 85 ff.

30 SCIENTIFIC METHOD IN PHILOSOPHY

Naturally enough, his followers divided into two
sects, the lovers of right-angled triangles and the
abhorrers of beans ; but the former sect died out,
leaving, however, a haunting flavour of mysticism
over much Greek mathematical speculation, and in
particular over Plato's views on mathematics. Plato,
of course, embodies both the scientific and mystical
attitudes in a higher form than his predecessors,
but the mystical attitude is distinctly the stronger
of the two, and secures ultimate victory whenever
the conflict is sharp. Plato, moreover, adopted from
the Eleatics the device of using logic to defeat common
sense, and thus to leave the field clear for mysticism
a device still employed in our own day by the adherents
of the classical tradition.

The logic used in defence of mysticism seems to me
faulty as logic, and in a later lecture I shall criticize
it on this ground. But the more thoroughgoing
mystics do not employ logic, which they despise:
they appeal instead directly to the immediate deliver-
ance of their insight. Now, although fully developed
mysticism is rare in the West, some tincture of it
colours the thoughts of many people, particularly as
regards matter on which they have strong convictions
not based on evidence. In all who seek passionately
for the fugitive and difficult goods, the conviction
is almost irresistible that there is in the world some-
thing deeper, more significant, than the multiplicity
of little facts chronicled and classified by science.
Behind the veil of these mundane things, they feel
something quite different obscurely shimmers, shining
forth clearly in the great moments of illumination,
which alone give anything worthy to be called real
knowledge of truth. To seek such moments, therefore,
is to them the way of wisdom, rather than, like the

CURRENT TENDENCIES 31

man of science, to observe coolly, to analyse without
emotion, and to accept without question the equal
reality of the trivial and the important.

Of the reality or unreality of the mystic's world I
know nothing. I have no wish to deny it, nor even
to declare that the insight which reveals it is not a
genuine insight. What I do wish to maintain
and it is here that the scientific attitude becomes
imperative is that insight, untested and unsupported,
is an insufficient guarantee of truth, in spite of the fact
that much of the most important truth is first sug-
gested by its means. It is common to speak of an
opposition between instinct and reason; in the
eighteenth century, the opposition was drawn in
favour of reason, but under the influence of Rousseau
and the romantic movement instinct was given the
preference, first by those who rebelled against arti-
ficial forms of government and thought, and then,
as the purely rationalistic defence of traditional theo-
logy became increasingly difficult, by all who felt in
science a menace to creeds which they associated
with a spiritual outlook on life and the world. Berg-
son, under the name of " intuition," has raised instinct
to the position of sole arbiter of metaphysical truth.
But in fact the opposition of instinct and reason is
mainly illusory. Instinct, intuition, or insight is
what first leads to the beliefs which subsequent reason
confirms or confutes ; but the confirmation, where it
is possible, consists, in the last analysis, of agreement
with other beliefs no less instinctive. Reason is a
harmonizing, controlling force rather than a creative
one. Even in the most purely logical realms, it is
insight that first arrives at what is new.

Where instinct and reason do sometimes conflict
is in regard to single beliefs, held instinctively, and

32 SCIENTIFIC METHOD IN PHILOSOPHY

held with such determination that no degree of incon-
sistency with other beliefs leads to their abandon-
ment. Instinct, like all human faculties, is liable to
error Those in whom reason is weak are often un-
willing to admit this as regards themselves, though
all admit it in regard to others. Where instinct is
least liable to error is in practical matters as to which
right judgment is a help to survival; friendship
and hostility in others, for instance, are often felt
with extraordinary discrimination through very care-
ful disguises. But even in such matters a wrong im-
pression may be given by reserve or flattery ; and
in matters less directly practical, such as philosophy
deals with, very strong instinctive beliefs may be
wholly mistaken, as we may come to know through
their perceived inconsistency with other equally
strong beliefs. It is such considerations that necessi-
tate the harmonizing mediation of reason, which
tests our beliefs by their mutual compatibility, and
examines, in doubtful cases, the possible sources of
error on the one side and on the other. In this there
is no opposition to instinct as a whole, but only to
blind reliance upon some one interesting aspect of
instinct to the exclusion of other more commonplace
but not less trustworthy aspects. It is such one-
sidedness, not instinct itself, that reason aims at
correcting.

These more or less trite maxims may be illustrated
by application to Bergson's advocacy of " intuition "
as against "intellect." There are, he says, "two
profoundly different ways of knowing a thing. The
first implies that we move round the object ; the
second that we enter into it. The first depends on
the point of view at which we are placed and on the
symbols by which we express ourselves. The second

CURRENT TENDENCIES 33

neither depends on a point of view nor relies on any
symbol. The first kind of knowledge may be said
to stop at the relative ; the second, in those cases
where it is possible, to attain the absolute" z The
second of these, which is intuition, is, he says, " the
kind of intellectual sympathy by which one places
oneself within an object in order to coincide with
what is unique in it and therefore inexpressible"
(p. 6). In illustration, he mentions self-knowledge :
" there is one reality, at least, which we all seize from
within, by intuition and not by simple analysis. It
is our own personality in its flowing through time
our self which endures " (p. 8). The rest of Bergson's
philosophy consists in reporting, through the imper-
fect medium of words, the knowledge gained by intui-
tion, and the consequent complete condemnation of
all the pretended knowledge derived from science and
common sense.

This procedure, since it takes sides in a conflict of
instinctive beliefs, stands in need of justification by
proving the greater trustworthiness of the beliefs on
one side than of those on the other. Bergson attempts
this justification in two ways first, by explaining that
intellect is a purely practical faculty designed to secure
biological success ; secondly, by mentioning remark-
able feats of instinct in animals, and by pointing out
characteristics of the world which, though intuition
can apprehend them, axe baffling to intellect as he
interprets it.

Of Bergson's theory that intellect is a purely prac-
tical faculty developed in the struggle for survival,
and not a source of true beliefs, we may say, first, that
it is only through intellect that we know of the struggle
for survival and of the biological ancestry of man : if
i Introduction to Metaphysics, p. i.
3

34 SCIENTIFIC METHOD IN PHILOSOPHY

the intellect is misleading, the whole of this merely
inferred history is presumably untrue. If, on the
other hand, we agree with M. Bergson in thinking that
evolution took place as Darwin believed, then it is
not only intellect, but all our faculties, that have been
developed under the stress of practical utility. In-
tuition is seen at its best where it is directly useful
for example, in regard to other people's characters
and dispositions. Bergson apparently holds that
capacity for this kind of knowledge is less explicable
by the struggle for existence than, for example,
capacity for pure mathematics. Yet the savage
deceived by false friendship is likely to pay for his
mistake with his life ; whereas even in the most
civilized societies men are not put to death for mathe-
matical incompetence. All the most striking of his
instances of intuition in animals have a very direct
survival value. The fact is, of course, that both in-
tuition and intellect have been developed because
they axe useful, and that, speaking broadly, they are
useful when they give truth and become harmful
when they give falsehood. Intellect, in civilized man,
like artistic capacity, has occasionally been developed
beyond the point where it is useful to the individual ;
intuition, on the other hand, seems on the whole to
diminish as civilization increases. Speaking broadly,
it is greater in children than in adults, in the un-
educated than in the educated. Probably in dogs it
exceeds anything to be found in human beings. But
those who find in these facts a recommendation of
intuition ought to return to running wild in the woods,
dyeing themselves with woad and living on hips and
haws.

Let us next examine whether intuition possesses any
such infaJlibiJity as Bergson claims for it. The best

CURRENT TENDENCIES 35

instance of it, according to him, is our acquaintance
with ourselves; yet self-knowledge is proverbially
rare and difficult. Most men, for example, have in
their nature meannesses, vanities, -and envies of which
they are quite unconscious, though even their best
friends can perceive them without any difficulty. It
is true that intuition has a convincingness which is
lacking to intellect : while it is present, it is almost
impossible to doubt its truth. But if it should appear,
on examination, to be at least as fallible as intellect,
its greater subjective certainty becomes a demerit,
making it only the more irresistibly deceptive. Apart
from self-knowledge, one of the most notable examples
of intuition is the knowledge people believe themselves
to possess of those with whom they are in love : the
wall between different personalities seems to become
transparent, and people flifriTr they see into another
soul as into their own. Yet deception in such cases
is constantly practised with success ; and even where
there is no intentional deception, experience gradually
proves, as a rule, that the supposed insight was illusory,
and that the slower, more groping methods of the
intellect are in the long run more reliable.

Bergson maintains that intellect can only deal
with things in so far as they resemble what has been
experienced in the past, while intuition has the power
of apprehending the uniqueness and novelty that
always belong to each fresh moment. That there is
something unique and new at every moment, is cer-
tainly true ; it is also true that this cannot be fully
expressed by means of intellectual concepts. Only
direct acquaintance can give knowledge of what is
unique and new. But direct acquaintance of this
kind is given fully in sensation, and does not require,
so far as I can see, any special faculty of intuition for

3 6 SCIENTIFIC METHOD IN PHILOSOPHY

its apprehension. It is neither intellect nor intuition,
but sensation, that supplies new data ; but when the
data are new in any remarkable manner, intellect is
much more capable of dealing with them than intui-
tion would be. The hen with a brood of ducklings
no doubt has intuitions which seem to place her inside
them, and not merely to know them analytically ;
but when the ducklings take to the water, the whole
apparent intuition is seen to be illusory, and the hen
is left helpless on the shore. Intuition, in fact, is an
aspect and development of instinct, and, like all
instinct, is admirable in those customary surroundings
which have moulded the habits of the animal in
question, but totally incompetent as soon as the
surroundings are changed in a way which demands
some non-habitual mode of action.

The theoretical understanding of the world, which
is the aim of philosophy, is not a matter of great
practical importance to animals, or to savages, or
even to most civilized men. It is hardly to be sup-
posed, therefore, that the rapid, rough and ready
methods of instinct or intuition will find in this field
a favourable ground for their application. It is the
older kinds of activity, which bring out our kinship
with remote generations of animal and semi-human
ancestors, that show intuition at its best. In such
matters as self-preservation and love, intuition will
act sometimes (though not always) with a swiftness
and precision which are astonishing to the critical
intellect. But philosophy is not one of the pursuits
which illustrate our affinity with the past : it is a
highly refined, highly civilized pursuit, demanding,
for its success, a certain liberation from the life of
instinct, and even, at times, a certain aloofness from
all mundane hopes and fears. It is not in philosophy,

CURRENT TENDENCIES 37

therefore, that we can hope to see intuition at its
best. On the contrary, since the true objects of
philosophy, and the habits of thought demanded for
their apprehension, are strange, unusual, and remote,
it is here, more almost than anywhere else, that in-
tellect proves superior to intuition, and that quick
unanalysed convictions are least deserving of uncritical
acceptance.

Before embarking upon the somewhat difficult and
abstract discussions which lie before us, it will be well
to take a survey of the hopes we may retain and the
hopes we must abandon. The hope of satisfaction
to our more human desires the hope of demonstrating
that the world has this or that desirable ethical charac-
teristic is not one which, so far as I can see, philosophy
can do anything whatever to satisfy. The difference
between a good world and a bad one is a difference
in the particular characteristics of the particular
things that exist in these worlds : it is not a sufficiently
abstract difference to come within the province of
philosophy. Love and hate, for example, are ethical
opposites, but to philosophy they are dosely analogous
attitudes towards objects. The general form and
structure of those attitudes towards objects which
constitute mental phenomena is a problem for philo-
sophy ; but the difference between love and hate is not
a difference of form or structure, and therefore belongs
rather to the special science of psychology than to
philosophy. Thus the ethical interests which have
often inspired philosophers must remain in the back-
ground : some kind of ethical interest may inspire
the whole study, but none must obtrude in the detail
or be expected in the special results which are sought.

If this view seems at first sight disappointing, we
may remind ourselves that a similar change has been

38 SCIENTIFIC METHOD IN PHILOSOPHY

found necessary in all the other sciences. The physi-
cist or chemist is not now required to prove the ethical
importance of his ions or atoms ; the biologist is not
expected to prove the utility of the plants or animals
which he dissects. In pre-scientific ages this was not
the case. Astronomy, for example, was studied
because men believed in astrology : it was thought
that the movements of the planets had the most direct
and important bearing upon the lives of human beings.
Presumably, when this belief decayed and the dis-
interested study of astronomy began, many who had
found astrology absorbingly interesting decided that
astronomy had too little human interest to be worthy
of study. Physics, as it appears in Plato's Timaus
for example, is full of ethical notions : it is an essential
part of its purpose to show that the earth is worthy
of admiration. The modern physicist, on the con-
trary, though he has no wish to deny that the earth
is admirable, is not concerned, as physicist, with its
ethical tributes : he is merely concerned to find out
facts, not to consider whether they are good or bad.
In psychology, the scientific attitude is even more
recent and more difficult than in the physical sciences :
it is natural to consider that human nature is either
good or bad, and to suppose that the difference between
good and bad, so all-important in practice, must be
important in theory also. It is only during the last
century that an ethically neutral science of psychology
has grown up; and here too ethical neutrality has
been essential to scientific success.

In philosophy, hitherto, ethical neutrality has been
seldom sought and hardly ever achieved. Men have
remembered their wishes, and have judged philosophies
in relation to their wishes. Driven from the par-
ticular sciences, the belief that the notions of good

CURRENT TENDENCIES 39

and evil must afford a key to the understanding of
the world has sought a refuge in philosophy. But
even from this last refuge, if philosophy is not to
remain a set of pleasing dreams, this belief must be
driven forth. It is a commonplace that happiness
is not best achieved by those who seek it directly ;
and it would seem that the same is true of the good.
In thought, at any rate, those who forget good and
evil and seek only to know the facts are more likely
to achieve good than those who view the world through
the distorting medium of their own desires.

The immense extension of our knowledge of facts
in recent times has had, as it had in the Renaissance,
two effects upon the general intellectual outlook.
On the one hand, it has made men distrustful of the
truth of wide, ambitious systems : theories come and
go swiftly, each serving, for a moment, to classify
known facts and promote the search for new ones,
but each in turn proving inadequate to deal with the
new facts when they have been found. Even those
who invent the theories do not, in science, regard them
as anything but a temporary makeshift. The ideal
of an all-embracing synthesis, such as the Middle
Ages believed themselves to have attained, recedes
further and further beyond the limits of what seems
feasible. In such a world, as in the world of Mon-
taigne, nothing seems worth while except the dis-
covery of more and more facts, each in turn the death-
blow to some cherished theory ; the ordering intellect
grows weary, and becomes slovenly through despair.

On the other hand, the new facts have brought new
powers; man's physical control over natural forces
has been increasing with unexampled rapidity, and
promises to increase in the future beyond all easily
assignable limits- ^ m alongside of despair ag

40 SCIENTIFIC METHOD IN PHILOSOPHY

regards ultimate theory there is an immense optimism
as regards practice : what man can do seems almost
boundless. The old fixed limits of human power,
such as death, or the dependence of the race on an
equilibrium of cosmic forces, are forgotten, and no
hard facts are allowed to break in upon the dream
of omnipotence. No philosophy is tolerated which
sets bounds to man's capacity of gratifying his wishes ;
and thus the very despair of theory is invoked to
silence every whisper of doubt as regards the possi-
bilities of practical achievement.

In the welcoming of new fact, and in the suspicion
of dogmatism as regards the universe at large, the
modern spirit should, I think, be accepted as wholly
an advance. But both in its practical pretensions
and in its theoretical despair it seems to me to go
too far. Most of what is greatest in man is called
forth in response to the thwarting of his hopes by
immutable natural obstacles; by the pretence of
omnipotence, he becomes trivial and a little absurd.
And on the theoretical side, ultimate metaphysical
truth, though less all-embracing and harder of attain-
ment than it appeared to some philosophers in the
past, can, I believe, be discovered by those who are
willing to combine the hopefulness, patience, and
open-mindedness of science with something of the
Greek feeling for beauty in the abstract world of
logic and for the ultimate intrinsic value in the con-
templation of truth.

The philosophy, therefore, which is to be genuinely
inspired by the scientific spirit, must deal with some-
what dry and abstract matters, and must not hope
to find an answer to the practical problems of life. To
those who wish to understand much of what has in
the past been most difficult and obscure in the constitu-

CURRENT TENDENCIES 41

tion of the universe, it has great rewards to offer
triumphs as noteworthy as those of Newton and
Darwin, and as important, in the long run, for the
moulding of our mental habits. And it brings with
it as a new and powerful method of investigation
always does a sense of power and a hope of progress
more reliable and better grounded than any that
rests on hasty and fallacious generalization as to the
nature of the universe at large. Many hopes which
inspired philosophers in the past it cannot da-im to
fulfil ; but other hopes, more purely intellectual, it
can satisfy more fully than former ages could have
deemed possible for human minds.

LECTURE II

LOGIC AS THE ESSENCE OF
PHILOSOPHY

THE topics we discussed in our first lecture, and the
topics we shall discuss later, all reduce themselves,
in so far as they are genuinely philosophical, to prob-
lems of logic. This is not due to any accident, but
to the fact that every philosophical problem, when it
is subjected to the necessary analysis and purification,
is found either to be not really philosophical at all,
or else to be, in the sense in which we are using the
word, logical. But as the word "logic" is never
used in the same sense by two different philosophers,
some explanation of what I mean by the word is
indispensable at the outset.

Logic, in the Middle Ages, and down to the present
day in teaching, meant no more than a scholastic
collection of technical terms and rules of syllogistic
inference. Aristotle had spoken, and it was the part
of humbler men merely to repeat the lesson after him.
The trivial nonsense embodied in this tradition is still
set in examinations, and defended by eminent authori-
ties as an excellent " propaedeutic," i.e. a training in
those habits of solemn humbug which are so great a
help in later life. But it is not this that I mean to
praise in saying that all philosophy is logic. Ever
since the beginning of the seventeenth centuiy, all

LOGIC AS THE ESSENCE OF PHILOSOPHY 43

vigorous minds that have concerned themselves with
inference have abandoned the mediaeval tradition, and
in one way or other have widened the scope of logic.

The first extension was the introduction of the
inductive method by Bacon and Galileo by the
former in a theoretical and largely mistaken form,
by the latter in actual use in establishing the founda-
tions of modern physics and astronomy. This is
probably the only extension of the old logic which has
become familiar to the general educated public. But
induction, important as it is when regarded as a method
of investigation, does not seem to remain when its
work is done : in the final form of a perfected science,
it would seem that everything ought to be deductive.
If induction remains at all, which is a difficult question,
it will remain merely as one of the principles according
to which deductions are effected. Thus the ultimate
result of the introduction of the inductive method
seems not the creation of a new kind of non-deductive
reasoning, but rather the widening of the scope of de-
duction by pointing out a way of deducing which is
certainly not syllogistic, and does not fit into the
mediaeval scheme.

The question of the scope and validity of induction
is of great difficulty, and of great importance to our
knowledge. Take such a question as, " Will the sun
rise to-morrow ? " Our first instinctive feeling is
that we have abundant reason for saying that it will,
because it has risen on so many previous mornings.
Now, I do not myself know whether this does afford
a ground or not, but I am willing to suppose that it
does. The question which then arises is : " What is
the principle of inference by which we pass from past
sunrises to future ones ? The answer given by Mill
is that the inference depends upon the law of causation.

44 SCIENTIFIC METHOD IN PHILOSOPHY

Let us suppose this to be true ; then what is the
reason for believing in the law of causation ? There
are broadly three possible answers : (i) that it is itself
known a priori ; (2) that it is a postulate ; (3) that
it is an empirical generalization from past instances
in which it has been found to hold. The theory that
causation is known a priori cannot be definitely refuted,
but it can be rendered very implausible by the mere
process of formulating the law exactly, and thereby
showing that it is immensely more complicated and
less obvious than is generally supposed. The theory
that causation is a postulate, i.e. that it is something
which we choose to assert although we know that it
is very likely false, is also incapable of refutation ; but
it is plainly also incapable of justifying any use of the
law in inference. We are thus brought to the theory
that the law is an empirical generalization, which is
the view held by Mill.

But if so, how are empirical generalizations to be
justified ? The evidence in their favour cannot be
empirical, since we wish to argue from what has been
observed to what has not been observed, which can only
be done by means of some known relation of the
observed and the unobserved; but the unobserved,
by definition, is not known empirically, and therefore
its relation to the observed, if known at all, must be
known independently of empirical evidence. Let us
see what Mill says on this subject.

According to Mill, the law of causation is proved by
an admittedly fallible process called "induction by
simple enumeration." This process, he says, " con-
sists in ascribing the nature of general truths to all
propositions which are true in every instance that we
happen to know of." J As regards its fallibility, he
* Logic, Book III. chapter ill. 2.

LOGIC AS THE ESSENCE OF PHILOSOPHY 45

asserts that r< the precariousness of the method of
simple enumeration is in an inverse ratio to the large-
ness of the generalization. The process is delusive and
insufficient, exactly in proportion as the subject-matter
of the observation is special and limited in extent.
As the sphere widens, this unscientific method becomes
less and less liable to mislead ; and the most universal
class of truths, the law of causation for instance, and
the principles of number and of geometry, are duly and
satisfactorily proved by that method alone, nor are
they susceptible of any other proof," x

In the above statement, there are two obvious
lacunae : (i) How is the method of simple enumeration
itself justified? (2) What logical principle, if any,
covers the same ground as this method, without
being liable to its failures ? Let us take the second
question first.

A method of proof which, when used as directed,
gives sometimes truth and sometimes falsehood as
the method of simple enumeration does is obviously
not a valid method, for validity demands invariable
truth. Thus, if simple enumeration is to be rendered
valid, it must not be stated as Mill states it. We shall
have to say, at most, that the data render the result
probable. Causation holds, we shall say, in every
instance we have been able to test ; therefore it probably
holds in untested instances. There are terrible diffi-
culties in the notion of probability, but we may ignore
them at present, We thus have what at least may
be a logical principte, since it is without exception.
If a proposition is true in every instance that we happen
to know of, and if the instances are very numerous,
then, we shall say, it becomes very probable, on the
data, that it will be true in any further instance. This
x Book III. chapter acri. 3.

46 SCIENTIFIC METHOD IN PHILOSOPHY

is not refuted by the fact that what we declare to be
probable does not always happen, for an event may be
probable on the data and yet not occur. It is, however,
obviously capable of further analysis, and of more
exact statement. We shall have to say something
like this : that every instance of a proposition z being
true increases the probability of its being true in a fresh
instance, and that a sufficient number of favourable
instances will, in the absence of instances to the contrary,
make the probability of the truth of a fresh instance
approach indefinitely near to certainty. Some such
principle as this is required if the method of simple
enumeration is to be valid.

But this brings us to our other question, namely,
how is our principle known to be true ? Obviously,
since it is required to justify induction, it cannot be
proved by induction ; since it goes beyond the empirical
data, it cannot be proved by them alone ; since it is
required to justify all inferences from empirical data
to what goes beyond them, it cannot itself be even
rendered in any degree probable by such data. Hence,
if it is known, it is not known by experience, but
independently of experience. I do not say that any
such principle is known : I only say that it is required
to justify the inferences from experience which empiri-
cists allow, and that it cannot itself be justified
empirically. 3

A similar conclusion can be proved by similar
arguments concerning any other logical principle.
Thus logical knowledge is not derivable from experi-
ence alone, and the empiricist's philosophy can
therefore not be accepted in its entirety, in spite

K Or rather a prepositional function.
* The subject of causality and induction will be discussed
again in Lecture VIII.

LOGIC AS THE ESSENCE OF PHILOSOPHY 4;

of its excellence in many matters which He outside
logic.

Hegel and his followers widened the scope of logic
in quite a different way a way which I believe to be
fallacious, but which requires discussion if only to show
how their conception of logic differs from the con-
ception which I wish to advocate. In their writings,
logic is practically identical with metaphysics. In
broad outline, the way this came about is as follows.
Hegel believed that, by means of a priori reasoning,
it could be shown that the world must have various
important and interesting characteristics, since any
world without these characteristics would be impossible
and self-contradictory. Thus what he calls "logic"
is an investigation of the nature of the universe, in so
far as this can be inferred merely from the principle
that the universe must be logically self-consistent.
I do not myself believe that from this principle alone
anything of importance can be inferred as regards the
existing universe. But, however that may be, I
should not regard Hegel's reasoning, even if it were
valid, as properly belonging to logic : it would rather
be an application of logic to the actual world. Logic
itself would be concerned rather with such questions
as what self-consistency is, which Hegel, so far as I
know, does not discuss. .And though he criticizes the
traditional logic, and professes to replace it by an
improved logic of his own, there is some sense in which
the traditional logic, with all its faults, is uncritically
and unconsciously assumed throughout his reasoning.
It is not in the direction advocated by him, it seems to
me, that the reform of logic is to be sought, but^by a
more fundamental, more patient, and less ambitious
investigation into the presuppositions which his system
shares with those of most other philosophers.

48 SCIENTIFIC METHOD IN PHILOSOPHY

The way in which, as it seems to me, Hegel's system
assumes the ordinary logic which it subsequently
criticizes, is exemplified by the general conception of
"categories" with which he operates throughout.
This conception is, I think, essentially a product of
logical confusion, but it seems in some way to stand
for the conception of " qualities of Reality as a whole."
Mr. Bradley has worked out a theory according to which,
in all judgment, we are ascribing a predicate to Reality
as a whole ; and this theory is derived from Hegel.
Now the traditional logic holds that every proposition
ascribes a predicate to a subject, and from this it easily
follows that there can be only one subject, the Absolute,
for if there were two, the proposition that there were two
would not ascribe a predicate to either. Thus Hegel's
doctrine, that philosophical propositions must be of
the form, " the Absolute is such-and-such," depends
upon the traditional belief in the universality of the
subject-predicate fonn. This belief, being traditional,
scarcely self-conscious, and not supposed to be impor-
tant, operates underground, and is assumed in argu-
ments which, like the refutation of relations, appear
at first sight such as to establish its truth. This is
the most important respect in which Hegel uncritically
assumes the traditional logic. Other less important
respects though important enough to be the source
of such essentially Hegelian conceptions as the " con-
crete universal " and the " union of identity in differ-
ence "will be found where he explicitly deals with
formal logic. 1

* See the translation by EL S. Macran, Hegel's Doctrine of
Formal Logic, Oxford, 1912. Hegel's argument in this
portion of bis " Logic " depends throughout upon confusing
the " is " of predication, as in " Socrates is mortal," with the
" is " of identity, as in " Socrates is the philosopher who drank

LOGIC AS THE ESSENCE OF PHILOSOPHY 49

There is quite another direction in which a large
technical development of logic has taken place : I
mean the dkection of what is called logistic or mathe-
matical logic This kind of logic is mathematical in
two different senses : it is itself a branch of mathe-
matics, and it is the logic which is specially applicable
to other more traditional branches of mathematics,
Historically, it began as merely a branch of mathematics:
its special applicability to other branches is a more
recent development. In both respects, it is the fulfil-
ment of a hope which Leibniz cherished throughout his
life, and pursued with all the ardour of his amazing
intellectual energy. Much of his work on this subject
has been published recently, since his discoveries have
been remade by others ; but none was published by
him, because his results persisted in contradicting
certain points in the traditional doctrine of the
syllogism. We now know that on these points the
traditional doctrine is wrong, but respect for Aristotle
prevented Leibniz from realizing that this was possible. 1

The modern development of mathematical logic

the hemlock. 1 ' Owing to tbfa confusion, he ^Tiinlrg that
" Socrates " and " mortal" must be identical. Seeing that
they are different, he does not infer, as others would, that there
is a mistake somewhere, but that they exhibit " identity in
difference." Again, Socrates is particular, "mortal" is
universal. Therefore, he says, since Socrates is mortal, it
follows that the particular is the universal taking the " is "
to be throughout expressive of identity. But to say " the
particular is the universal" is self -contradictory. Again
Hegel does not suspect a mistake but proceeds to synthesize
particular and universal in the individual, or concrete universal.
This is an example of how, for want of care at the start, vast
and imposing systems of philosophy axe built upon stupid and
trivial confusions, which, but for the almost incredible fact
that they are unintentional, one would be tempted to charac-.
terize as puns.

* Cf. Couturat, La Logique de Leibniz, pp. 361, 386.

4

50 SCIENTIFIC METHOD IN PHILOSOPHY

dates from Boole's Laws of Thought (1854). But in
him and his successors, before Peano and Frege, the
only thing really achieved, apart from certain details,
was the invention of a mathematical symbolism for
deducing consequences from the premisses which the
newer methods shared with those of Aristotle. This
subject has considerable interest as an independent
branch of mathematics, but it has very little to do with
real logic. The first serious advance in real logic since
the time of the Greeks was made independently by
Peano and Frege both mathematicians. They both
arrived at their logical results by an analysis of mathe-
matics. Traditional logic regarded the two propositions,
" Socrates is mortal " and " All men are mortal," as
being of the same form ; * Peano and Frege showed
that they are utterly different in form. The philosophical
importance of logic may be illustrated by the fact that
this confusion which is still committed by most
writers obscured not only the whole study of the
forms of judgment and inference, but also the relations
of things to their qualities, of concrete existence to
abstract concepts, and of the world of sense to the world
of Platonic ideas. Peano and Frege, who pointed out
the error, did so for technical reasons, and applied their
logic mainly to technical developments ; but the
philosophical importance of the advance which they
made is impossible to exaggerate.

Mathematical logic, even in its most modern form,
is not directly of philosophical importance except
in its beginnings. After the beginnings, it belongs
rather to mathematics than to philosophy. Of its
beginnings, which are the only part of it that can

* It was often recognized that there was some difference
between them, but it was not recognized that the difference
is fundamental, and of very great importance.

LOGIC AS THE ESSENCE OF PHILOSOPHY 51

properly be called philosophical logic, I shall speak
shortly. But even the later developments, though
not directly philosophical, will be f ound of great indirect
use in philosophizing. They enable us to deal easily
with more abstract conceptions than merely verbal
reasoning can enumerate ; they suggest fruitful hypo-
theses which otherwise could hardly be thought of ;
and they enable us to see quickly what is the smallest
store of materials with which a given logical or scientific
edifice can be constructed. Not only Frege's theory
of number, which we shall deal with in Lecture VII,
but the whole theory of physical concepts which will
be outlined in our next two lectures, is inspired by
mathematical logic, and could never have been
imagined without it.

In both these cases, and in many others, we shall
appeal to a certain principle called " the principle of
abstraction." This principle, which might equally well
be called " the principle which dispenses with abstrac-
tion," and is one which clears away incredible accumu-
lations of metaphysical lumber, was directly suggested
by mathematical logic, and could hardly have been
proved or practically used without its help. The
principle will be explained in our fourth lecture, but
its use may be briefly indicated in advance. When a
group of objects have that kind of similarity which
we are inclined to attribute to possession of a common
quality, the principle in question shows that membership
of the group will serve all the purposes of the supposed
common quality, and that therefore, unless some
common quality is actually known, the group or class
of gin-nlar objects may be used to replace the common
quality, which need not be assumed to exist. In this
and other ways, the indirect uses of even the later parts
of mathematical logic are very great ; but it is now

52 SCIENTIFIC METHOD IN PHILOSOPHY

time to turn our attention to its philosophical founda-
tions.

In every proposition and in every inference there
is, besides the particular subject-matter concerned,
a certain form, a way in which the constituents of the
proposition or inference are put together. If I say,
" Socrates is mortal," " Jones is angry," " The sun is
hot," there is something in common in these three
cases, something indicated by the word " is." What
is in common is the form of the proposition, not an
actual constituent. If I say a number of things about
Socrates that he was an Athenian, that he married
Xantippe, that he drank the hemlock there is a
common constituent, namely Socrates, in all the propo-
sitions I enunciate, but they have diverse forms. If,
on the other hand, I take any one of these propositions
and replace its constituents, one at a time, by other
constituents, the form remains constant, but no con-
stituent remains. Take (say) the series of propositions,
"Socrates drank the hemlock," "Coleridge drank
the hemlock/' " Coleridge drank opium," " Coleridge
ate opium." The form remains unchanged throughout
this series, but all the constituents are altered. Thus
form is not another constituent, but is the way the
constituents axe put together. It is forms, in this
sense, that are the proper object of philosophical
logic.

It is obvious that the knowledge of logical forms
is something quite different from knowledge of existing
things. The form of " Socrates drank the hemlock "
is not an existing thing like Socrates or the hemlock,
nor does it even have that dose relation to existing
things that drinking has. It is something altogether
more abstract and remote. We might understand all
the separate words of a sentence without understanding

LOGIC AS THE ESSENCE OF PHILOSOPHY 53

the sentence : if a sentence is long and complicated,
this is apt to happen. In such a case we have knowledge
of the constituents, but not of the form. We may also
have knowledge of the f orm without having knowledge
of the constituents. If I say, " Rorarius drank the
hemlock/ 1 those among you who have never heard of
Rorarius (supposing there are any) will understand the
form, without having knowledge of all the constituents.
In order to understand a sentence, it is necessary to
have knowledge berth of the constituents and of the
particular instance of the form. It is in this way that
a sentence conveys iaf onnation, since it tells us that
certain known objects are related according to a certain
known form. Thus some kind of knowledge of logical
forms, though with most people it is not explicit, is
involved in all understanding of discourse. It is the
business of philosophical logic to extract this knowledge
from its concrete integuments, and to render it explicit
and pure.

In all inference, form alone is essential : the particu-
lar subject-matter is irrelevant except as securing the
truth of the premisses. This is one reason for the
great importance of logical form. When I say,
" Socrates was a man, all men are mortal, therefore
Socrates was mortal," the connection of premisses
and conclusion does not in any way depend upon its
being Socrates and man and mortality that I am
mentioning. The general form of the inference may be
expressed in some such words as : " If a thing has a
certain property, and whatever has this property has
a certain other property, then the thing in question
also has that other property." Here no particular
things or properties axe mentioned : the proposition
is absolutely general. All inferences, when stated
fully, are instances of propositions having this kind of

54 SCIENTIFIC METHOD IN PHILOSOPHY

generality. If they seem to depend upon the subject-
matter otherwise than as regards the truth of the
premisses, that is because the premisses have not been
all explicitly stated. In logic, it is a waste of time to
deal with inferences concerning particular cases :
we deal throughout with completely general and purely
formal implications, leaving it to other sciences to
discover when the hypotheses axe verified and when
they are not.

But the forms of propositions giving rise to inferences
are not the simplest forms ; they are always hypo-
thetical, stating that if one proposition is true, then
so is another. Before considering inference, there-
fore, logic must consider those simpler forms which
inference presiipposes. Here the traditional logic
failed completely: it believed that there was only
one form of simple proposition (i.e. of proposition
not stating a relation between two or more other
propositions), namely, the form which ascribes a
predicate to a subject. This is the appropriate form
in assigning the qualities of a given thing we may
say " this thing is round, and red, and so on." Gram-
mar favours this form, but philosophically it is so far
from universal that it is not even very common. If
we say " this thing is bigger than that," we are not
assigning a mere quality of " this," but a relation of
"this" and "that." We might express the same
fact by saying " that thing is smaller than this," where
grammatically the subject is changed. Thus propo-
sitions stating that two things have a certain relation
have a different form from subject-predicate propo-
sitions, and the failure to perceive this difference or
to allow for it has been the source of many errors in
traditional metaphysics.

The belief or unconscious conviction that all propo

LOGIC AS THE ESSENCE OF PHILOSOPHY 55

sitions are of the subject-predicate form in other
words : that every fact consists in some thing having
some quality has rendered most philosophers incapable
of giving any account of the world of science and daily
life. If they had been honestly anxious to give such
an account, they would probably have discovered
their error very quickly ; but most of them were less
anxious to understand the world of science and daily
life, than to convict it of unreality in the interests
of a super-sensible "real" world. Belief in the
unreality of the world of sense arises with irresistible
force in certain moods moods which, I imagine, have
some simple physiological basis, but are none the
less powerfully persuasive. The conviction born of
these moods is the source of most mysticism and
of most metaphysics. When the emotional intensity of
such a mood subsides, a man who is in the habit of
reasoning will search for logical reasons in favour
of the belief which he finds in himself. But since the
belief already exists, he will be very hospitable to any
reason that suggests itself. The paradoxes apparently
proved by his logic are really the paradoxes of mysticism,
and are the goal which he feds his logic must reach
if it is to be in accordance with insight. It is in this
way that logic has been pursued by those of the great
philosophers who were mystics notably Plato, Spinoza,
and Hegel. But since they usually took for granted
the supposed insight of the mystic emotion, their
logical doctrines were presented with a certain dryness,
and were believed by their disciples to be quite inde-
pendent of the sudden jl.nnTnTjna.fifm from which they
sprang. Nevertheless their origin dung to them, and
they remained to borrow a useful word from Mr.
Santayana " malicious " in regard to the world of
science and common sense. It is only so that we

56 SCIENTIFIC METHOD IN PHILOSOPHY

can account for the complacency with which philo-
sophers have accepted the inconsistence of their
doctrines with all the common and scientific facts
which seem best established and most worthy of belief.

The logic of mysticism shows, as is natural, the
defects which are inherent in anything malicious.
While the mystic mood is dominant, the need of logic
is not felt ; as the mood fades, the impulse to logic
reasserts itself, but with a desire to retain the vanishing
insight, or at least to prove that it was insight, and
that what seems to contradict it is illusion. The logic
which thus arises is not quite disinterested or candid,
and is inspired by a certain hatred of the daily world
to which it is to be applied. Such an attitude naturally
does not tend to the best results. Everyone knows
that to read an author simply in order to refute him
is not the way to understand him ; and to read the
book of Nature with a conviction that it is all illusion
is just as unlikely to lead to understanding. If our
logic is to find the common world intelligible, it must
not be hostile, but must be inspired by a genuine
acceptance such as is not usually to be found among
metaphysicians.

Traditional logic, since it holds that all propositions
have the subject-predicate form, is unable to admit
the reality of relations : all relations, it maintains,
must be reduced to properties of the apparently related
terms. There are many ways of refuting this opinion ;
one of the easiest is derived from the consideration
of what are called " asymmetrical " relations. In
order to explain this, I will first explain two independent
ways of classifying relations

Some relations, when they hold between A and B,
also hold between B and A. Such, for example, is
the relation " brother or sister." If A is a brother or

LOGIC AS THE ESSENCE OF PHILOSOPHY 57

sister of B thsn B is a brother or sister of A. Such
again is any kind of similarity, say similarity of colour.
Any kind of dissimilarity is also of this kind : if the
colour of A is unlike the colour of B, then the colour of
B is unlike the colour of A. Relations of this sort are
called symmetrical. Thus a relation is symmetrical
if, whenever it holds between A and B, it also holds
between B and A.

All relations that are not symmetrical are called
non-symmetrical. Thus " brother " is ncni-syimnetrical,
because, if A is a brother of B, it may happen that
B is a sister of A.

A relation is called asymmetrical when, if it holds
between A and B, it never holds between B and A.
Thus husband, father, grandfather, etc., are asym-
metrical relations. So are before, after, greater, above,
to the right of, etc. All the relations that give rise to
series are of this kind.

Classification into symmetrical, asymmetrical and
merely non-symmetrical relations is the first of the
two classifications we had to consider. The second
is into transitive, intransitive, and merely non-transitive
relations, which are defined as follows.

A relation is said to be transitive, if, whenever it
holds between A and B and also between B and C,
it holds between A and C. Thus before, after, greater,
above are transitive, All relations giving rise to series
are transitive, but so are many others. The transitive
relations just mentioned were asymmetrical, but
many transitive relations are symmetrical f or instance,
equality in any respect, exact identity of colour, being
equally numerous (as applied to collections), and
so on.

A relation is said to be non-transitive whenever it
is not transitive. Thus " brother " is non-transitive,

58 SCIENTIFIC METHOD IN PHILOSOPHY

because a brother of one's brother may be oneself.
All kinds of dissimilarity are non-transitive.

A relation is said to be intransitive when, if A has the
relation to B, and B to C, A never has it to C. Thus
" father " is intransitive. So is such a relation as
" one inch taller " or " one year later."

Let us now, in the light of this classification, return
to the question whether all relations can be reduced
to predications.

In the case of symmetrical relations i.e. relations
which, if they hold between A and B, also hold between
B and A some kind of plausibility can be given to
this doctrine. A symmetrical relation which is
transitive, such as equality, can be regarded as expres-
sing possession of some common property, while one
which is not transitive, such as inequality, can be
regarded as expressing possession of different properties.
But when we come to asymmetrical relations, such as
before and after, greater and less, etc., the attempt
to reduce them to properties becomes obviously
impossible. When, for example, two things are merely
known to be unequal, without our knowing which
is greater, we may say that the inequality results
from their having different magnitudes, because
inequality is a symmetrical relation ; but to say that
when one thing is greater than another, and not merely
unequal to it, that means that they have different
magnitudes, is formally incapable of explaining the
facts. For if the other thing had been greater than
the one, the magnitudes would also have been different,
though the fact to be explained would not have been
the same. Thus mere difference of magnitude is not all
that is involved, since, if it were, there would be no
difference between one thing being greater than another,
and the other being greater than the one. We shall

LOGIC AS THE ESSENCE OF PHILOSOPHY 59

have to say that the one magnitude is greater than
the other , and thus we shall have failed to get rid of
the relation " greater." In short, both possession of
the same property and possession of different properties
are symmetrical relations, and therefore cannot account
for the existence of asymmetrical relations.

Asymmetrical relations are involved in all series
in space and time, greater and less, whole and part,
and many others of the most important characteristics
of the actual world. All these aspects, therefore, the
logic which reduces everything to subjects and predi-
cates is compelled to condemn as error and mere
appearance. To those whose logic is not malicious,
such a wholesale condemnation appears impossible.
And in fact there is no reason except prejudice, so far
as I can discover, for denying the reality of relations.
When once their reality is admitted, all kgical grounds
for supposing the world of sense to be illusory disappear.
If this is to be supposed, it must be frankly and simply
on the ground of mystic insight unsupported by
argument. It is impossible to argue against what
professes to be insight, so long as it does not argue in
its own favour. As logicians, therefore, we may
admit the possibility of the mystic's world, while yet,
so long as we do not have his insight, we must continue
to study the everyday world with which we are
familiar. But when he contends that our world is
impossible, then our logic is ready to repel his attack.
And the first step in creating the logic which is to
perform this service is the recognition of the reality of
relations.

Relations which have two terms are only one kind
of relations. A relation may have three terms, or four,
or any number. Relations of two terms, being the
simplest, have received more attention than the

60 SCIENTIFIC METHOD IN PHILOSOPHY

others, and have generally been alone considered by
philosophers, both those who accepted and those
who denied the reality of relations. But other relations
have their importance, and are indispensable in the
solution of certain problems. Jealousy, for example,
is a relation between three people. Professor Royce
mentions the relation " giving " : when A gives B
to C, that is a relation of three terms. 1 When a man
says to his wife : " My dear, I wish you could induce
Angelina to accept Edwin," his wish constitutes a
relation between four people, himself, his wife, Angelina,
and Edwin. Thus such relations are by no means
recondite or rare. But in order to explain exactly
how they differ from relations of two terms, we must
embark upon a classification of the logical forms of
facts, which is the first business of logic, and the
business in which the traditional logic has been most
deficient.

The existing world consists of many things with
many qualities and relations. A complete description
of the existing world would require not only a catalogue
of the things, but also a mention of all their qualities
and relations. We should have to know not only this
that, and the other thing, but also which was red,
which yellow, which was earlier than which, which was
which between two others, and so on. When I speak
of a " fact," I do not mean one of the simple things
in the world ; I mean that a certain thing has a certain
quality, or that certain things have a certain relation.
Thus, for example, I should not call Napoleon a fact,
but I should call it a fact that he was ambitious, or
that he married Josephine. Now a fact, in this sense,
is never simple, but always has two or more constitu-
ents. When it simply assigns a quality to a thing,

Encyclopedia of the Philosophical Sciences, vol. i. p. 97.

LOGIC AS THE ESSENCE OF PHILOSOPHY 61

it has only two constituents, the thing and the quality.
When it consists of a relation between two things,
it has three constituents, the things and the relation.
When it consists of a relation between three things,
it has four constituents, and so on. The constituents
of facts, in the sense in which we are using the
word " fact," are not other facts, but are things
and qualities or relations. When we say that there
are relations of more than two terms, we mean that
there are single facts consisting of a single relation
and more than two things. I do not mean that
one relation of two terms may hold between A and
B, and also between A and C, as, for example, a
man is the son of his father and also the son of his
mother. This constitutes two distinct facts: if we
choose to treat it as one fact, it is a fact which has
facts for its constituents. But the facts I am speaking
of have no facts among their constituents, but only
things and relations. For example, when A is jealous
of B on account of C, there is only one fact, involving
three people ; there are not two instances of jealousy,
but only one. It is in such cases that I speak of a
relation of three terms, where the simplest possible
fact in which the relation occurs is one involving three
things in addition to the relation. And the same
applies to relations of four terms or five or any other
number. All such relations must be admitted in
our inventory of the logical forms of facts : two facts
involving the same number of things have the same
form, and two which involve different numbers of
things have different forms.

Given any fact, there is an assertion which expresses
the fact. The fact itself is objective, and independent
of our thought or opinion about it ; but the assertion
is something which involves thought, and may be

62 SCIENTIFIC METHOD IN PHILOSOPHY

either true or false. An assertion may be positive or
negative : we may assert that Charles I was executed,
or that he did not die in his bed. A negative assertion
may be said to be a denial. Given a form of words
which must be either true or false, such as " Charles I
died in his bed," we may either assert or deny this
form of words : in the one case we have a positive
assertion, in the other a negative one. A form of
words which must be either true or false I shall call
a proposition. Thus a proposition is the same as what
may be significantly asserted or denied. A proposition
which expresses what we have called a fact, i.e. which,
when asserted, asserts that a certain thing has a
certain quality, or that certain things have a certain
relation, will be called an atomic proposition, because,
as we shall see immediately, there are other propositions
into which atomic propositions enter in a way analogous
to that in which atoms enter into molecules. Atomic
propositions, although, like facts, they may have any
one of an infinite number of forms, are only one kind
of propositions. All other kinds are more complicated.
In order to preserve the parallelism in language
as regards facts and propositions, we shall give the name
" atomic facts " to the facts we have hitherto been
considering. Thus atomic facts are what determine
whether atomic propositions are to be asserted or
denied.

Whether an atomic proposition, such as "this is
red," or "this is before that," is to be asserted or
denied can only be known empirically. Perhaps one
atomic fact may sometimes be capable of being inferred
from another, though this seems very doubtful ; but
in any case it cannot be inferred from premisses no
one of which is an atomic fact. It follows that, if
atomic facts are to be known at all, some at least must

LOGIC AS THE ESSENCE OF PHILOSOPHY 63

be known without inference. The atomic facts which
we come to know in this way are the facts of sense-
perception; at any rate, the facts of sense-percep-
tion are those which we most obviously and
certainly come to know in this way. If we knew all
atomic facts, and also knew that there were none
except those we knew, we should, theoretically, be
able to infer all truths of whatever form. 1 Thus logic
would then supply us with the whole of the apparatus
required. But in the first acquisition of knowledge
concerning atomic facts, logic is useless. In pure logic,
no atomic fact is ever mentioned : we confine ourselves
whofly to forms, without asking ourselves what objects
can fill the forms. Thus pure logic is independent of
atomic facts ; but conversely, they are, in a sense,
independent of logic. Pure logic and atomic facts
are the two poles, the wholly a priori and the wholly
empirical. But between the two lies a vast intermediate
region, which we must now briefly explore.

" Molecular " propositions are such as contain con-
junctions if, or, and, unless, etc. and such words
are the marks of a molecular proposition. Consider
such an assertion as, " If it rains, I shall bring my
umbrella." This assertion is just as capable of truth
or falsehood as the assertion of an atomic proposition,
but it is obvious that either the corresponding fact,
or the nature of the correspondence with fact, must
be quite different from what it is in the case of an atomic
proposition. Whether it rains, and whether I bring
my umbrella, are each severally matters of atomic

* This perhaps requires modification in order to include
such facts as beliefs and wishes, since such facts apparently
contain propositions as components. Such facts, though not
strictly atomic, must be supposed included if the statement
in the text is to be true.

64 SCIENTIFIC METHOD IN PHILOSOPHY

fact, ascertainable by observation. But the connection
of the two involved in saying that if the one happens,
then the other will happen, is something radically
different from either of the two separately. It does not
require for its truth that it should actually rain, or that
I should actually bring my umbrella; even if the
weather is cloudless, it may still be true that I should
have brought my umbrella if the weather had been
different. Thus we have here a connection of two
propositions, which does not depend upon whether
they are to be asserted or denied, but only upon the
second being inferable from the first. Such propositions,
therefore, have a f oim which is different from that of
any atomic proposition.

Such propositions are important to logic, because all
inference depends upon them. If I have told you that if
it rains I shall bring my umbrella, and if you see that
there is a steady downpour, you can infer that I shall
bring my umbrella. There can be no inference except
where propositions are connected in some such way,
so that from the truth or falsehood of the one something
follows as to the truth or falsehood of the other. It
seems to be the case that we can sometimes know
molecular propositions, as in the above instance of
the umbrella, when we do not know whether the
component atomic propositions are true or false. The
practical utility of inference rests upon this fact.

The next kind of propositions we have to consider
are general propositions, such as " all men are mortal/'
" all equilateral triangles are equiangular." And with
these belong propositions in which the word " some "
occurs, such as " some men are philosophers " or " some
philosophers are not wise." These are the denials of
general propositions, namely (in the above instances),
of " all men are non-philosophers " and " all philoso-

LOGIC AS THE ESSENCE OF PHILOSOPHY 65

pliers are wise." We will call propositions containing the
word" some " negative general propositions, and those
containing the word " all " positive general propositions.
These propositions, it will be seen, begin to have the
appearance of the propositions in logical text-books.
But their peculiarity and complexity are not known
to the text-books, and the problems which they raise
are only discussed in the most superficial ma.nTip.r-

When we were discussing atomic facts, we saw that
we should be able, theoretically, to infer all other
truths by logic if we knew all atomic facts and also knew
that there were no other atomic facts besides those we
knew. The knowledge that there are no other atomic
facts is positive general knowledge ; it is the knowledge
that " all atomic facts are known to me," or at least
" all atomic facts are in this collection "however the
collection may be given. It is easy to see that general
propositions, such as " all men are mortal," cannot be
known by inference from atomic facts alone. If we
could know each individual man, and know that he
was mortal, that would not enable us to know that all
men are mortal, unless we knew that those were all the
men there are, which is a general proposition. If we
knew every other existing thing throughout the universe,
and knew that each separate thing was not an immortal
man, that would not give us our result unless we knew
that we had explored the whole universe, i.e. unless
we knew " all things belong to this collection of things
I have examined." Thus general truths cannot be
inferred from particular truths alone, but must, if
they are to be known, be either self-evident or inferred
from premisses of which at least one is a general truth.
But all empirical evidence is of particular truths.
Hence, if there is any knowledge of general truths at
all, there must be some knowledge of general truths

5

66 SCIENTIFIC METHOD IN PHILOSOPHY

which is independent of empirical evidence, i.e. does
not depend upon the data of sense.

The above conclusion, of which we had an instance
in the case of the inductive principle, is important,
since it affords a refutation of the older empiricists.
They believed that all our knowledge is derived from
the senses and dependent upon them. We see that,
if this view is to be maintained, we must refuse to admit
that we know any general propositions. It is perfectly
possible logically that this should be the case, but it
does not appear to be so in fact, and indeed no one
would dream of maintaining such a view except a
theorist at the last extremity. We must therefore
admit that there is general knowledge not derived from
sense, and that some of this knowledge is not obtained
by inference but is primitive.

Such general knowledge is to be found in logic.
Whether there is any such knowledge not derived
from logic, I do not know ; but in logic, at any rate,
we have such knowledge. It will be remembered that
we excluded from pure logic such propositions as,
"Socrates is a man, all men are mortal, therefore
Socrates is mortal," because Socrates and man and
mortal are empirical terms, only to be understood
through particular experience. The corresponding
proposition in pure logic is : " If anything has a
certain property, and whatever has this property
has a certain other property, then the thing in
question has the other property." This proposition is
absolutely general: it applies to all things and all
properties. And it is quite self-evident. Thus in such
propositions of pure logic we have the self-evident
general propositions of which we were in search.

A proposition such as " If Socrates is a man, and all
men are mortal, then Socrates is mortal," is true in

LOGIC AS THE ESSENCE OF PHILOSOPHY 67
virtue of its form alone. Its truth, in this hypothetical
form, does not depend upon whether Socrates actually
is a man, nor upon whether in fact all men are mortal ;
thus it is equally true when we substitute other terms
for Socrates and man and mortal. The general truth
of which it is an instance is purely formal, and belongs
to logic. Since this general truth does not mention
any particular thing, or even any particular quality
or relation, it is wholly independent of the accidental
facts of the existent world, and can be known, theo-
retically, without any experience of particular things
or their qualities and relations.

Logic, we may say, consists of two parts. The first
part investigates what propositions are and what
forms they may have; this part enumerates the
different kinds of atomic propositions, of molecular
propositions, of general propositions, and so on. The
second part consists of certain supremely general
propositions, which assert the truth of all propositions
of certain forms. This second part merges into pure
mathematics, whose propositions all turn out, on
analysis, to be such general formal truths. The first
part, which merely enumerates forms, is the more
difficult, and philosophically the more important;
and it is the recent progress in this first part, more
than anything else, that has rendered a truly scientific
discussion of many philosophical problems possible.

The problem of the nature of judgment or belief
may be taken as an example of a problem whose
solution depends upon an adequate inventory of logical
forms. We have already seen how the supposed
universality of the subject-predicate form made it
impossible to give a right analysis of serial order, and
therefore made space and time unintelligible. But in
this case it was only necessary to admit relations of

68 SCIENTIFIC METHOD IN PHILOSOPHY

two terms. The case of judgment demands the admis-
sion of more complicated forms. If all judgments were
true, we might suppose that a judgment consisted in
apprehension of zfact, and that the apprehension was
a relation of a mind to the fact. From poverty in the
logical inventory, this view has often been held. But
it leads to absolutely insoluble difficulties in the case
of error. Suppose I believe that Charles I died in his
bed. There is no objective fact " Charles I's death in
his bed" to which I can have a relation of appre-
hension. Charles I and death and his bed are objective,
but they are not, except in my thought, put together
as my false belief supposes. It is therefore necessary,
in analysing a belief, to look for some other logical
form than a two-term relation. Failure to realize this
necessity has, in my opinion, vitiated almost everything
that has hitherto been written on the theory of know-
ledge, making the problem of error insoluble and the
difference between belief and perception inexplicable.

Modern logic, as I hope is now evident, has the effect
of enlarging our abstract imagination, and providing
an infinite number of possible hypotheses to be applied
in the analysis of any complex fact. In this respect
it is the exact opposite of the logic practised by the
classical tradition. In that logic, hypotheses which
seem prima fade possible are professedly proved
impossible, and it is decreed in advance that reality
must have a certain special character. In modern
logic, on the contrary, while the prima fade hypotheses
as a rule remain admissible, others, which only logic
would have suggested, axe added to our stock, and are
very often found to be indispensable if a right analysis
of the facts is to be obtained. The old logic put thought
in fetters, while the new logic gives it wings. It has,
in my opinion, introduced the same kind of advance

LOGIC AS THE ESSENCE OF PHILOSOPHY 69

into philosophy as Galileo introduced into physics,
making it possible at last to see what kinds of problems
may be capable of solution, and what kinds must be
abandoned as beyond human powers. And where a
solution appears possible, the new logic provides a
method which enables us to obtain, results that do not
merely embody personal idiosyncrasies, but must
command the assent of all who are competent to form
an opinion.

LECTURE III

ON OUR KNOWLEDGE OF THE EXTERNAL
WORLD

PHILOSOPHY may be approached by many roads, but
one of .the oldest and most travelled is the road which
leads through doubt as to the reality of the world of
sense. In Indian mysticism, in Greek and modern
monistic philosophy from Parmenides onward, in
Berkeley, in modern physics, we find sensible appear-
ance criticized and condemned for a bewildering
variety of motives. The mystic condemns it on the
ground of immediate knowledge of a more real and
significant world behind the veil ; Parmenides and
Plato condemn it because its continual flux is thought
inconsistent with the unchanging nature of the abstract
entities revealed by logical analysis ; Berkeley brings
several weapons, but his chief is the subjectivity of
sense-data, their dependence upon the organization and
point of view of the spectator ; while modern physics,
on the basis of sensible evidence itself, maintains a
mad dance of electrons which have, superficiaJly ( at
least, very little resemblance to the immediate objects
of sight or touch.

Every one of these lines of attack raises vital and
interesting problems.

The mystic, so long as he merely reports a positive
revelation, cannot be refuted; but when he denies

THE EXTERNAL WORLD 71

reality to objects of sense, he may be questioned as to
what he means by " reality," and may be asked how
their unreality follows from the supposed reality of his
super-sensible world. In answering these questions,
he is led to a logic which merges into that of Parraenid.es
and Plato and the idealist tradition.

The logic of the idealist tradition has gradually grown
very complex and very abstruse, as may be seen from
the Bradleian sample considered in our first lecture.
If we attempted to deal fully with this logic, we should
not have time to reach any other aspect of our subject ;
we will therefore, while acknowledging that it deserves
a long discussion, pass by its central doctrines with
only such occasional criticism as may serve to exemplify
other topics, and concentrate our attention on such
matters as its objections to the continuity of motion
and the infinity of space and time objections which
have been fully answered by modern mathematicians
in a manner constituting an abiding triumph for the
method of logical analysis in philosophy. These
objections and the modern answers to them will occupy
our fifth, sixth, and seventh lectures.

Berkeley's attack, as reinforced by the physiology of
the sense-organs and nerves and brain, is very powerful.
I think it must be admitted as probable that the imme-
diate objects of sense depend for their existence upon
physiological conditions in ourselves, and that, for
example, the coloured surfaces which we see cease to
exist when we shut our eyes. But it would be a mistake
to infer that they are dependent upon mind, not real
while we see them, or not the sole basis for our know-
ledge of the external world. This line of argument
will be developed in the present lecture.

The discrepancy between the world of physics and
the world of sense, which we shall consider in our

72 SCIENTIFIC METHOD IN PHILOSOPHY

fourth lecture, will be found to be more apparent
than real, and it will be shown that whatever there
is reason to believe in physics can probably be inter-
preted consistently with the reality of sense-data.

The instrument of discovery throughout is modern
logic, a very different science from the logic of the
text-books and also from the logic of idealism. Our
second lecture has given a short account of modern
logic and of its points of divergence from the various
traditional kinds of logic.

In our last lecture, after a discussion of causality
and free will, we shall try to reach a general account
of the logical-analytic method of scientific philosophy,
and a tentative estimate of the hopes of philosophical
progress which it allows us to entertain.

In this lecture, I wish to apply the logical-analytic
method to one of the oldest problems of philosophy,
namely, the problem of our knowledge of the external
world. What I have to say on this problem does not
amount to an answer of a definite and dogmatic kind ;
it amounts only to an analysis and statement of the
questions involved, with an indication of the directions
in which evidence may be sought. But although not
yet a definite solution, what can be said at present
seems to me to throw a completely new light on the
problem, and to be indispensable, not only in seeking
the answer, but also in the preliminary question as
to what parts of our problem may possibly have an
ascertainable answer.

In every philosophical problem, our investigation
starts from what may be called " data," by which I
mean matters of common knowledge, vague, complex,
inexact, as common knowledge always is, but yet
somehow commanding our assent as on the whole and
in some interpretation pretty certainly true. In the

THE EXTERNAL WORLD 73

case of our present problem, the common knowledge
involved is of various kinds. There is first our acquain-
tance with particular objects of daily life furniture,
houses, towns, other people, and so on. Then there
is the extension of such particular knowledge to par-
ticular things outside our personal experience, through
history and geography, newspapers, etc. And lastly,
there is the systematization of all this knowledge of
particulars by means of physical science, which derives
immftngft persuasive force from its astonishing power
of foretelling the future. We are quite willing to
admit that there may be errors of detail in this know-
ledge, but we believe them to be discoverable and
corrigible by the methods which have given rise to our
beliefs, and we do not, as practical men, entertain
for a moment the hypothesis that the whole edifice
may be built on insecure foundations. In the main,
therefore, and without absolute dogmatism as to this
or that special portion, we may accept this mass of
common knowledge as affording data for our philo-
sophical analysis.

It may be said and this is an objection which must
be met at the outset that it is the duty of the philo-
sopher to call in question the admittedly fallible beliefs
of daily life, and to replace them by something more
solid and irrefragable. In a sense this is true, and in
a sense it is effected in the course of analysis. But
in another sense, and a very important one, it is quite
impossible. While admitting that doubt is possible
with regard to all our common knowledge, we must
nevertheless accept that knowledge in the main if
philosophy is to be possible at alL There is not any
superfine brand of knowledge, obtainable by the
philosopher, which can give us a standpoint from which
to criticize the whole of the knowledge of daily life.

74 SCIENTIFIC METHOD IN PHILOSOPHY

The most that can be done is to examine and purify
our common knowledge by an internal scrutiny,
assuming the canons by which it has been obtained,
and applying them with more care and with more
precision. Philosophy cannot hoajsljolha^ing. achieved
such adggrfeof certjdnty jthat itj^_have authority"
' and the laiws'of

sceptical in regard to every detail, is not sceptical as
regards the whole. That is to say, its criticism of details
will only be based upon their relation to other details,
not upon some external criterion which can be applied
to all the details equally. The reason for this absten-
tion from a universal criticism is not any dogmatic
confidence, but its exact opposite; it is not that
common knowledge must be true, but that we possess
no radically different kind of knowledge derived from
some other source. Universal scepticism, though
logically irrefutable, is practically barren ; it can only,
therefore, give a certain flavour of hesitancy to our
beliefs, and cannot be used to substitute other beliefs
for them.

Although data can only be criticized by other data,
not by an outside standard, yet we may . distinguish
different grades of certainty in the different kinds of
common knowledge which we enumerated just now.
What does not go beyond our own personal sensible
acquaintance must be for us the most certain : the
" evidence of the senses " is proverbially the least
open to question. What depends on testimony, like
the facts of history and geography which are learnt
from books, has varying degrees of certainty according
to the nature and extent of the testimony. Doubts
as to the existence of Napoleon can only be maintained
for a joke, whereas the historicity of Agamemnon is

THE EXTERNAL WORLD 75

a legitimate subject of debate. In science, again, we
find all grades of certainty short of the highest. The
law of gravitation, at least as an approximate truth,
has acquired by this time the same kind of certainty
as the existence of Napoleon, whereas the latest specu-
lations concerning the constitution of matter would
be universally acknowledged to have as yet only a
rather slight probability in their favour. These varying
degrees of certainty attaching to different data may
be regarded as themselves forming part of our data ;
they, along with the other data, lie within the vague,
complex, inexact body of knowledge which it is the
business of the philosopher to analyse.

The first thing that appears when we begin to analyse
our common knowledge is that some of it is derivative,
while some is primitive ; that is to say, there is some
that we only believe because of something else from
which it has been inferred in some sense, though not
necessarily in a strict logical sense, while other parts
are believed on their own account, without the support
of any outside evidence. It is obvious that the senses
give knowledge of the latter kind: the immediate
facts perceived by sight or touch or hearing do not
need to be proved by argument, but are completely
self-evident. Psychologists, however, have made us
aware that what is actually given in sense is much
less than most people would naturally suppose, and
that much of what at first sight seems to be given is
really inferred. This applies especially in regard to our
space-perceptions. For instance, we unconsciously infer
the " real " size and shape of a visible object from its
apparent size and shape, according to its distance and
our point of view. When we hear a person speaking,
our actual sensations usually miss a great deal of what
he says, and we supply its place by unconscious

76 SCIENTIFIC METHOD IN PHILOSOPHY

inference ; in a foreign language, where this process
is more difficult, we find ourselves apparently grown
deaf, requiring, for example, to be much nearer the
stage at a theatre than would be necessary in our
own country. Thus the first step in the analysis of
data, namely, the discovery of what is really given in
sense, is full of difficulty. We will, however, not
linger on this point ; so long as its existence is realized,
the exact outcome does not make any very great difier-
ence in our main problem.

The next step in our analysis must be the con-
sideration of how the derivative parts of our common
knowledge arise. Here we become involved in a some-
what puzzling entanglement of logic and psychology.
Psychologically, a belief may be called derivative
whenever it is caused by one or more other beliefs,
or by some fact of sense which is not simply what the
belief asserts. Derivative beliefs in this sense con-
stantly arise without any process of logical inference,
merely by association of ideas or some equally extra-
logical process. From the expression of a man's face
we judge as to what he is feeling : we say we see that
he is angry, when in fact we only see a frown. We do
not judge as to his state of mind by any logical process :
the judgment grows up, often without our being able
to say what physical mark of emotion we actually
saw. In such a case, the knowledge is derivative
psychologically ; but logically it is in a sense primitive,
since it is not the result of any logical deduction.
There may or may not be a possible deduction leading
to the same result, but whether there is or not, we
certainly do not employ it. If we call a belief " logically
primitive" when it is not actually arrived at by a
logical inference, then innumerable beliefs are logically
primitive which psychologically are derivative. The

THE EXTERNAL WORLD 77

separation of these two kinds of primitiveness is vitally
important to our present discussion.

When we reflect upon the beliefs which axe logically
but not psychologically primitive, we find that, unless
they can on reflection be deduced by a logical process
from beliefs which are also psychologically primitive,
our confidence in their truth tends to diminish the more
we think about them. We naturally believe, for
example, that tables and chairs, trees and mountains,
are still there when we turn our backs upon them. I
do not wish for a moment to maintain that this is
certainly not the case, but I do maintain that the
question whether it is the case is not to be settled off-
hand on any supposed ground of obviousness. The
belief that they persist is, in all men except a few
philosophers, logically primitive, but it is not psycho-
logically primitive; psychologically, it arises only
through our having seen those tables and chairs, trees
and mountains. As soon as the question is seriously
raised whether, because we have seen them, we have a
right to suppose that they are there still, we feel that
some kind of argument must be produced, and that if
none is forthcoming, our belief can be no more than
a pious opinion. We do not feel this as regards the
immediate objects of sense : there they are, and as
far as their momentary existence is concerned, no
further argument is required. There is accordingly
more need of justifying our psychologically derivative
beliefs than of justifying those that are primitive.

We are thus led to a somewhat vague distinction
between what we may call " hard " data and " soft "
data. This distinction is a matter of degree, and must
not be pressed ; but if not taken too seriously, it may
help to make the situation clear. I mean by " hard "
data those which resist the solvent influence of critical

78 SCIENTIFIC METHOD IN PHILOSOPHY

reflection, and by " soft " data those which, under the
operation of this process, become to our minds more
or less doubtful. The hardest of hard data are of two
sorts : the particular facts of sense, and the general
truths of logic. The more we reflect upon these, the
more we realize exactly what they are, and exactly
what a doubt concerning them really means, the more
luminously certain do they become. Verbal doubt
concerning even these is possible, but verbal doubt
may occur when what is nominally being doubted is
not really in our thoughts, and only words are actually
present to our minds. Real doubt, in these two cases,
would, I think, be pathological. At any rate, to me
they seem quite certain, and I shall assume that you
agree with me in this. Without this assumption, we
are in danger of falling into that universal scepticism
which, as we saw, is as barren as it is irrefutable.
If we are to continue philosophizing, we must make
our bow to the sceptical hypothesis, and, while
admitting the elegant terseness of its philosophy,
proceed to the consideration of other hypotheses
which, though perhaps not certain, have at least as
good a right to our respect as the hypothesis of the
sceptic.

Applying our distinction of "hard" and "soft"
data to psychologically derivative but logically primi-
tive beliefs, we shall find that most, if not all, are to be
classed as soft data. They may be found, on reflection,
to be capable of logical proof, and they then again
become believed, but no longer as data. As data,
though entitled to a certain limited respect, they cannot
be placed on a level with the facts of sense or the laws
of logic. The kind of respect which they deserve
seems to me such as to warrant us in hoping, though
not too confidently, that the hard data may prove them

THE EXTERNAL WORLD 79

to be at least probable. Also, if the hard data are
found to throw no light whatever upon their truth or
falsehood, we are justified, I think, in giving rather
more weight to the hypothesis of their truth than to
the hypothesis of their falsehood. For the present,
however, let us confine ourselves to the hard data,
with a view to discovering what sort of world can be
constructed by their means alone.

Our data now are primarily the facts of sense (i.e.
of our own sense-data) and the laws of logic. But even
the severest scrutiny will allow some additions to this
slender stock. Some facts of memory especially
of recent memory seem to have the highest degree of
certainty. Some introspective facts are as certain as
any facts of sense. And facts of sense themselves must,
for our present purposes, be interpreted with a certain
latitude. Spatial and temporal relations must some-
times be included, for example in the case of a swift
motion falling wholly within the specious present.
And some facts of comparison, such as the likeness
or unlikeness of two shades of colour, are certainly
to be included among hard data. Also we must remem-
ber that the distinction of hard and soft data is psycho-
logical and subjective, so that, if there are other
minds than our own which at our present stage must
be held doubtful the catalogue of hard data may be
different for them from what it is for us.

Certain common beliefs are undoubtedly excluded
from hard data. Such is the belief which led us to
introduce the distinction, namely, that sensible objects
in general persist when we are not perceiving them.
Such also is the belief in other people's minds : this
belief is psychologically derivative from our perception
of their bodies, and is fdt to demand logical justifica-
tion as soon as we become aware of its derivativeness.

8o SCIENTIFIC METHOD IN PHILOSOPHY

Belief in what is reported by the testimony of others,
including all that we learn from books, is of course
involved in the doubt as to whether other people
have minds at all. Thus the world from which our
reconstruction is to begin is very fragmentary. The
best we can say for it is that it is slightly more extensive
than the world at which Descartes arrived by a similar
process, since that world contained nothing except
himself and his thoughts.

We are now in a position to understand and state
the problem of our knowledge of the external world,
and to remove various misunderstandings which have
obscured the meaning of the problem. The problem
really is : Can the existence of anything other
than our own hard data be inferred from the
existence of those data? But before considering
this problem, let us briefly consider what the problem
is not.

When we speak of the " external " world in this
discussion, we must not mean " spatially external,"
unless " space " is interpreted in a peculiar and recon-
dite manner. The immediate objects of sight, the
coloured surfaces which make up the visible world,
are spatially external in the natural meaning of this
phrase. We fed them to be " there " as opposed to
" here " ; without making any assumption of an
existence other than hard data, we can more or less
estimate the distance of a coloured surface. It seems
probable that distances, provided they are not too
great, are actually given more or less roughly in sight ;
but whether this is the case or not, ordinary distances
can certainly be estimated approximately by means
of the data of sense alone. The immediately given
world is spatial, and is further not wholly contained
within our own bodies, at least in the obvious sense.

THE EXTERNAL WORLD 81

Thus our knowledge of what is external in this sense
is not open to doubt.

Another form in which the question is often put is :
" Can we know of the existence of any reality which is
independent of ourselves ? " This form of the question
suffers from the ambiguity of the two words " inde-
pendent " and " self." To take the Self first : the
question as to what is to be reckoned part of the Self
and what is not, is a very difficult one. Among many
other things which we may mean by the Self, two may
be selected as specially important, namely (i) the bare
subject which thinks and is aware of objects, (2) the
whole assemblage of things that would necessarily
cease to exist if our lives came to an end. The bare
subject, if it exists at all, is an inference, and is not
part of the data; therefore, this meaning of Self
may be ignored in our present inquiry. The second
meaning is difficult to make precise, since we hardly
know what things depend upon our lives for their
existence. And in this form, the definition of Self
introduces the word " depend," which raises the same
questions as are raised by the word " independent."
Let us therefore take up the word " independent,"
and return to the Self later.

When we say that one thing is " independent "
of another, we may mean either that it is logically
possible for the one to exist without the other, or that
there is no causal relation between the two such that
the one only occurs as the effect of the other. The
only way, so far as I know, in which one thing can be
logically dependent upon another is when the other
is part of the one. The existence of a book, for example,
is logically dependent upon that of its pages : without
the pages there would be no book. Thus in this sense
the question, " Can we know of the existence of any

6

82 SCIENTIFIC METHOD IN PHILOSOPHY

reality which is independent of ourselves ? " reduces
to the question, " Can we know of the existence of
any reality of which our Self is not part ? " In this
form, the question brings us back to the problem of
defining the Self ; but I think, however the Self may be
defined, even when it is taken as the bare subject,
it cannot be supposed to be part of the immediate
object of sense ; thus in this form of the question we
must admit that we can know of the existence of
realities independent of ourselves.

The question of causal dependence is much more
difficult. To know that one kind of thing is causally
independent of another, we must know that it actually
occurs without the other. Now it is fairly obvious
that, whatever legitimate meaning we give to the Self,
our thoughts and feelings are causally dependent
upon ourselves, i.e. do not occur when there is no
Self for them to belong to. But in the case of objects
of sense this is not obvious ; indeed, as we saw, the
common-sense view is that such objects persist in the
absence of any percipient. If this is the case, then they
are causally independent of ourselves; if not, not.
Thus in this form the question reduces to the question
whether we can know that objects of sense, or any other
objects not our own thoughts and feelings, exist at
times when we are not perceiving them. This form,
in which the difficult word " independent " no longer
occurs, is the form in which we stated the problem a
minute ago.

Our question in the above form raises two distinct
problems, which it is important to keep separate.
First, can we know that objects of sense, or very
similar objects, exist at times when we are not perceiv-
ing them ? Secondly, if this cannot be known, can
we know that other objects, inferable from objects

THE EXTERNAL WORLD 83

of sense but not necessarily resembling them, exist
either when we are perceiving the objects of sense
or at any other time? This latter problem arises
in philosophy as the problem of the " thing in itself,"
and in science as the problem of matter as assumed
in physics. We will consider this latter problem
first.

According to some authors among whom I was
formerly included it is necessary to distinguish
between a sensation, which is a mental event, and its
object, which is a patch of colour or a noise or what
not. If this distinction is made, the object of the
sensation is called a " sense-datum " or a " sensible
object." Nothing in the problems to be discussed in
this book depends upon the question whether this
distinction is valid or not. If it is not valid, the sensa-
tion and the sense-datum are identical If it is valid,
it is the sense-datum which concerns us in this book,
not the sensation. For reasons explained in The Analy-
sis of Mind (e.g. p. 141 ff .) I have come to regard the
distinction as not valid, and to consider the sense-
datum identical with the sensation. But it will not
be necessary to assume the correctness of this view
in what follows.

When I speak of a " sensible object," it must be
understood that I do not mean such a thing as a table,
which is both visible and tangible, can be seen by
many people at once, and is more or less permanent.
What I mean is just that patch of colour which is
momentarily seen when we look at the table, or just
that particular hardness which is felt when we press
it, or just that particular sound which is heard when
we rap it. Both the thing-in-itself of philosophy
and the matter of physics present themselves as causes
of the sensible object as much as of the sensation

84 SCIENTIFIC METHOD IN PHILOSOPHY

(if these are distinct). What are the common grounds
for this opinion ?

In each case, I think, the opinion has resulted from
the combination of a belief that something which can
persist independently of our consciousness makes itself
known in sensation, with the fact that our sensations
often change in ways which seem to depend upon us
rather than upon anything which would be supposed
to persist independently of us. At first, we believe
unreflectingly that everything is as it seems to be,
and that, if we shut our eyes, the objects we had been
seeing remain as they were though we no longer see
them. But there are arguments against this view,
which have generally been thought conclusive. It
is extraordinarily difficult to see just what the
arguments prove; but if we are to make any
progress with the problem of the external world,
we must try to make up our minds as to these
arguments.

A table viewed from one place presents a different
appearance from that which it presents from another
place. This is the language of common sense, but
this language already assumes that there is a real table
of which we see the appearances. Let us try to state
what is known in terms of sensible objects alone,
without any element of hypothesis. We find that as we
walk round the table, we perceive a series of gradually
changing visible objects. But in speaking of " walking
round the table," we have still retained the hypothesis
that there is a single table connected with all the
appearances. What we ought to say is that, while
we have those muscular and other sensations which
make us say we are walking, our visual sensations
change in a continuous way, so that, for example,
a striking patch of colour is not suddenly replaced by

THE EXTERNAL WORLD 85

something wholly different, but is replaced by an
insensible gradation of slightly different colours with
slightly different shapes. This is what we really know
by experience, when we have freed our minds from
the assumption of permanent " things " with changing
appearances. What is really known is a correlation of
muscular and other bodily sensations with changes
in visual sensations.

But walking round the table is not the only way of
altering its appearance. We can shut one eye, or put
on blue spectacles, or look through a microscope.
All these operations, in various ways, alter the visual
appearance which we call that of the table. More
distant objects will also alter their .appearance if (as
we say) the state of the atmosphere changes if there
is fog or rain or sunshine. Physiological changes also
alter the appearances of things. If we assume the
world of common sense, all these changes, including
those attributed to physiological causes, are changes
in the intervening medium. It is not quite so easy as in
the former case to reduce this set of facts to a form
in which nothing is assumed beyond sensible objects.
Anything intervening between ourselves and what we
see must be invisible : our view in every direction is
bounded by the nearest visible object. It might
be objected that a dirty pane of glass, for example,
is visible, although we can see things through it. But
in this case we really see a spotted patchwork : the
dirtier specks in the glass are visible, while the cleaner
parts are invisible and allow us to see what is beyond.
Thus the discovery that the intervening medium affects
the appearances of things cannot be made by means of
the sense of sight alone.

Let us take the case of the blue spectacles, which is
the simplest, but may serve as a type for the others.

86 SCIENTIFIC METHOD IN PHILOSOPHY

The frame of the spectacles is of course visible, but the
blue glass, if it is dean, is not visible. The blueness,
which we say is in the glass, appears as being in the
objects seen through the glass. The glass itself is known
by means of the sense of touch. In order to know that
it is between us and the objects seen through it, we
must know how to correlate the space of touch with
the space of sight. This correlation itself, when stated
in terms of the data of sense alone, is by no means a
simple matter. But it presents no difficulties of
principle, and may therefore be supposed accomplished.
When it has been accomplished, it becomes possible to
attach a meaning to the statement that the blue glass,
which we can touch, is between us and the object seen,
as we say, " through " it.

But we have still not reduced our statement com*
pletdy to what is actually given in sense. We have
fallen into the assumption that the object of which
we are conscious when we touch the blue spectacles
still exists after we have ceased to touch them. So
long as we are touching them, nothing except our
finger can be seen through the part touched, which is
the only part where we immediately know that there
is something. If we are to account for the blue appear-
ance of objects other than the spectacles, when seen
through them, it might seem as if we must assume that
the spectacles still exist when we are not touching
them ; and if this assumption really is necessary, our
main problem is answered : we have means of knowing
of the present existence of objects not given in sense,
though of the same kind as objects formerly given
in sense.

It may be questioned, however, whether this assump-
tion is actually unavoidable, though it is unquestionably
the most natural one to make. We may say that the

THE EXTERNAL WORLD 87

object of which we become aware when we touch the
spectacles continues to have effects afterwards, though
perhaps it no longer exists. In this view, the supposed
continued existence of sensible objects after they have
ceased to be sensible will be a fallacious inference
from the fact that they still have effects. It is often
supposed that nothing which has ceased to exist can
continue to have effects, but this is a mere preju-
dice, due to a wrong conception of causality. We
cannot, therefore, dismiss our present hypothesis
on the ground of a priori impossibility, but must
examine further whether it can really account for
the facts.

It may be said that our hypothesis is useless in the
case when the blue glass is never touched at all. How,
in that case, are we to account for the blue appearance
of objects ? And more generally, what are we to make
of the hypothetical sensations of touch which we
associate with untouched visible objects, which we
know would be verified if we chose, though in fact we
do not verify them ? Must not these be attributed to
permanent possession, by the objects, of the properties
which touch would reveal ?

Let us consider the more general question first.
Experience has taught us that where we see certain
kinds of coloured surfaces we can, by touch, obtain
certain expected sensations of hardness or softness,
tactile shape, and so on. This leads us to believe that
what is seen is usually tangible, and that it has, whether
we touch it or not, the hardness or softness which we
should expect to fed if we touched it. But the mere
fact that we are able to infer what our tactile sensations
would be shows that it is not logically necessary to
assume tactile qualities before they are felt. All that
is really known is that the visual appearance in question

88 SCIENTIFIC METHOD IN PHILOSOPHY

together with touch, will lead to certain sensations,
which can necessarily be determined in terms of the
visual appearance, since otherwise they could not be
inferred from it.

We can now give a statement of the experienced
facts concerning the blue spectacles, which will supply
an interpretation of common-sense beliefs without
assuming anything beyond the existence of sensible
objects at the times when they are sensible. By
experience of the correlation of touch and sight sensa-
tions, we become able to associate a certain place in
touch-space with a certain corresponding place in
sight-space. Sometimes, namely in the case of trans-
parent things, we find that there is a tangible object
in a touch-place without there being any visible object
in the corresponding sight-place. But in such a case
as that of the blue spectacles, we find that whatever
object is visible beyond the empty sight-place in the
same line of sight has a different colour from what it
has when there is no tangible object in the intervening
touch-place ; and as we move the tangible object in
touch-space, the blue patch moves in sight-space. If
now we find a blue patch moving in this way in sight-
space, when we have no sensible experience of an
intervening tangible object, we nevertheless infer that,
if we put our hand at a certain place in touch-space,
we should experience a certain touch-sensation. If we
are to avoid non-sensible objects, this must be taken
as the whole of our meaning when we say that the
blue spectacles are in a certain place, though we have
not touched them, and have only seen other things
rendered blue by their interposition.

I think it may be laid down quite generally that,
in so far as physics or common sense is verifiable, it
must be capable of interpretation in terms of actual

THE EXTERNAL WORLD 89

sense-data alone. The reason for this is simple.
Verification consists always in the occurrence of an
expected sense-datum. Astronomers tell us there
will be an eclipse of the moon : we look at the moon,
and find the earth's shadow biting into it, that is to
say, we see an appearance quite different from that
of the usual full moon. Now if an expected sense-
datum constitutes a verification, what was asserted
must have been about sense-data; or, at any rate,
if part of what was asserted was not about sense-data,
then only the other part has been verified. There is
in fact a certain regularity or conformity to law about
the occurrence of sense-data, but the sense-data that
occur at one time are often causally connected with
those that occur at quite other times, and not, or
at least not very closely, with those that occur at
neighbouring times. If I look at the moon and imme-
diately afterwards hear a train coming, there is no very
close causal connection between my two sense-data ;
but if I look at the moon on two nights a week apart,
there is a very dose causal connection between the two
sense-data. The simplest, or at least the easiest,
statement of the connection is obtained by imagining
a " real " moon which goes on whether I look at it
or not, providing a series of possible sense-data of
which only those are actual which belongs to moments
when I choose to look at the moon.

But the degree of verification obtainable in this way
is very small. It must be remembered that, at our
present level of doubt, we are not at liberty to accept
testimony. When we hear certain noises, which are
those we should utter if we wished to express a certain
thought, we assume that that thought, or one very
like it, has been in another mind, and has given rise
to the expression which we hear. If at the same time

90 SCIENTIFIC METHOD IN PHILOSOPHY

we see a body resembling our own, moving its lips
as we move ours when we speak, we cannot resist the
belief that it is alive, and that the feelings inside it
continue when we are not looking at it. When we
see our friend drop a weight upon his toe, and hear
him say what we should say in similar circumstances,
the phenomena can no doubt be explained without
assuming that he is anything but a series of shapes
and noises seen and heard by us, but practically no
man is so infected with philosophy as not to be quite
certain that his friend has felt the same kind of pain as
he himself would feel. We will consider the legitimacy
of this belief presently ; for the moment, I only wish
to point out that it needs the same kind of justification
as our belief that the moon exists when we do not see
it, and that, without it, testimony heard or read is
reduced to noises and shapes, and cannot be regarded
as evidence of the facts which it reports. The verifica-
tion of physics which is possible at our present level
is, therefore, only that degree of verification which is
possible by one man's unaided observations, which
will not carry us very far towards the establishment
of a whole science.

Before proceeding further, let us summarize the
argument so far as it has gone. The problem is : " Can
the existence of anything other than our own hard
data be inferred from these data ? " It is a mistake
to state the problem in the form : " Can we know of
the existence of anything other than ourselves and
our states ? " or : " Can we know of the existence of
anything independent of ourselves ? " because of the
extreme difficulty of defining "self" and "inde-
pendent " precisely. The felt passivity of sensation
is irrelevant, since, even if it proved anything, it
could only prove that sensations are caused by sensible

THE EXTERNAL WORLD 91

objects. The natural naive belief is that things seen
persist, when unseen, exactly or approximately as
they appeared when seen ; but this belief tends to be
dispelled by the fact that what common sense regards
as the appearance of one object changes with what
common sense regards as changes in the point of
view and in the intervening medium, including in the
latter our own sense-organs and nerves and brain.
This fact, as just stated, assumes, however, the common-
sense world of stable objects which it professes to call
in question ; hence, before we can discover its precise
bearing on our problem, we must find a way of stating
it which does not involve any of the assumptions
which it is designed to render doubtful. What we
then find, as the bare outcome of experience, is that
gradual changes in certain sense-data are correlated
with gradual changes in certain others, or (in the
case of bodily motions) with the other sense-data
themselves.

The assumption that sensible objects persist after
they have ceased to be sensible for example, that
the hardness of a visible body, which has been dis-
covered by touch, continues when the body is no longer
touched may be replaced by the statement that the
effects of sensible objects persist, i.e. that what happens
now can only be accounted for, in many cases, by
taking account of what happened at an earlier time.
Everything that one man, by his own personal experi-
ence, can verify in the account of the world given by
common sense and physics, will be explicable by some
such means, since verification consists merely in the
occurrence of an expected sense-datum. But what
depends upon testimony, whether heard or read, cannot
be explained in this way, since testimony depends
upon the existence of minds other than our own, and

92 SCIENTIFIC METHOD IN PHILOSOPHY

thus requires a knowledge of something not given in
sense. But before examining the question of our
knowledge of other minds, let us return to the question
of the thing-in-itself, namely, to the theory that what
exists at times when we axe not perceiving a given
sensible object is something quite unlike that object,
something which, together with us and our sense-
organs, causes our sensations, but is never itself given
in sensation.

The thing-in-itself, when we start from common-
sense assumptions, is a fairly natural outcome of the
difficulties due to the changing appearances of what
is supposed to be one object. It is supposed that the
table (for example) causes our sense-data of sight and
touch, but must, since these are altered by the point
of view and the intervening medium, be quite different
from the sense-data to which it gives rise. The objection
to this theory, I think, lies in its failure to realize
the radical nature of the reconstruction demanded by
the difficulties to which it points. We cannot speak
legitimately of changes in the point of view and the
intervening medium until we have already constructed
some world more stable than that of momentary
sensation. Our discussion of the blue spectacles and
the walk round the table has, I hope, made this dear.
But what remains far from dear is the nature of the
reconstruction required.

Although we cannot rest content with the above
theory, in the terms in which it is stated, we must
neverthdess treat it with a certain respect, for it is
in outline the theory upon which physical science and
physiology are built, and it must, therefore, be suscep-
tible of a true interpretation. Let us see how this is
to be done.

The first thing to realize is that there are no such

THE EXTERNAL WORLD 93

things as " illusions of sense." Objects of sense, even
when they occur in dreams, are the most indubitably
real objects known to us. What, then, makes us call
them unreal in dreams ? Merely the unusual nature
of their connection with other objects of sense. I dream
that I am in America, but I wake up and find myself
in England without those intervening days on the
Atlantic which, alas 1 are inseparably connected with
a "real" visit to America. Objects of sense are
called " real " when they have the kind of connection
with other objects of sense which experience has led us
to regard as normal ; when they fail in this, they are
called " illusions." But what is illusory is only the
inferences to which they give rise ; in themselves, they
are every bit as real as the objects of waking life.
And conversely, the sensible objects of waking life
must not be expected to have any more intrinsic reality
than those of dreams. Dreams and waking life, in our
first efforts at construction, must be treated with equal
respect ; it is only by some reality not merely sensible
that dreams can be condemned.

Accepting the indubitable momentary reality of
objects of sense, the next thing to notice is the
confusion underlying objections derived from their
changeableness. As we walk round the table, its
aspect changes ; but it is thought impossible to maintain
either that the table changes, or that its various
aspects can all " really " exist in the same place. If
we press one eyeball, we shall see two tables ; but it
is thought preposterous to maintain that there are
"really" two tables. Such arguments, however,
seem to involve the assumption that there can be
something more real than objects of sense. If we
see two tables, then there are two visual tables. It
is perfectly true that, at the same moment, we may

94 SCIENTIFIC METHOD IN PHILOSOPHY

discover by touch that there is only one tactile table.
This makes us declare the two visual tables an illusion,
because usually one visual object corresponds to one
tactile object. But all that we are warranted in saying
is that, in this case, the manner of correlation of touch
and sight is unusual. Again, when the aspect of the
table changes as we walk round it, and we are told
there cannot be so many different aspects in the same
place, the answer is simple : what does the critic of
the table mean by " the same place " ? The use of
such a phrase presupposes that all our difficulties have
been solved ; as yet, we have no right to speak of a
"place" except with reference to one given set of
momentary sense-data. When all are changed by a
bodily movement, no place remains the same as it
was. Thus the difficulty, if it exists, has at least not
been rightly stated.

We will now make a new start, adopting a different
method. Instead of inquiring what is the minimum of
assumption by which we can explain the world of sense,
we will, in order to have a model hypothesis as a
help for the imagination, construct one possible
(not necessary) explanation of the facts. It may
perhaps then be possible to pare away what is
superfluous in our hypothesis, leaving a residue
which may be regarded as the abstract answer to our
problem.

Let us imagine that each mind looks out upon the
world, as in Leibniz's monadology, from a point of
view peculiar to itself ; and for the sake of simplicity
let us confine ourselves to the sense of sight, ignoring
minds which are devoid of this sense. Each mind sees
at each moment an immensely complex three-dimen-
sional world ; but there is absolutely nothing which
is seen by two minds simultaneously. When we say

THE EXTERNAL WORLD 95

that two people see the same thing, we always find that,
owing to difference of point of view, there are differences,
however slight, between their immediate sensible
objects. (I am here assuming the validity of testimony
but as we are only constructing a possible theory, that
is a legitimate assumption.) The three-dimensional
world seen by one mind therefore contains no place
in common with that seen by another, for places can
only be constituted by the things in or around them.
Hence we may suppose, in spite of the differences
between the different worlds, that each exists entire
exactly as it is perceived, and might be exactly as it
is even if it were not perceived. We may further
suppose that there are an infinite number of such
worlds which are in fact unperceived. If two men are
sitting in a room, two somewhat similar worlds are
perceived by them ; if a third man enters and sits
between them, a third world, intermediate between
the two previous worlds, begins to be perceived.
It is true that we cannot reasonably suppose just this
world to have existed before, because it is conditioned
by the sense-organs, nerves, and brain of the newly
arrived man; but we can reasonably suppose that
some aspect of the universe existed from that point of
view, though no one was perceiving it. The system
consisting of all views of the universe, perceived and
unperceived, I shall call the system of "perspectives" ;
I shall confine the expression " private worlds " to
such views of the universe as are actually perceived.
Thus a " private world " is a perceived "perspective "
but there may be any number of unperceived per-
spectives.

Two men are sometimes found to perceive very
similar perspectives, so similar that they can use the
same words to describe them. They say they see

96 SCIENTIFIC METHOD IN PHILOSOPHY

the same table, because the differences between the two
tables they see are slight and not practically important.
Thus it is possible, sometimes, to establish a correlation
by similarity between a great many of the things of
one perspective, and a great many of the things of
another. In case the similarity is very great, we say
the points of view of the two perspectives are near
together in space-; but this space in which they are
near together is totally different from the spaces
inside the two perspectives. It is a relation between the
perspectives, and is not in either of them ; no one can
perceive it, and if it is to be known it can be only by
inference. Between two perceived perspectives which
are similar, we can imagine a whole series of other
perspectives, some at least unperceived, and such
that between any two, however similar, there are others
still more similar. In this way the space which consists
of relations between perspectives can be rendered
continuous, and (if we choose) three-dimensional.

We can now define the momentary common-sense
"thing," as opposed to its momentary appearances.
By the similarity of neighbouring perspectives, many
objects in the one can be correlated with objects in
the other, namely with the similar objects. Given an
object in one perspective, form the system of all the
objects correlated with it in all the perspectives ; that
system may be identified with the momentary com-
mon-sense " thing." Thus an aspect of a "thing" is a
member of the system of aspects which is the " thing "
at that moment. (The correlation of the times of
different perspectives raises certain complications, of
the kind considered in the theory of relativity ; but
we may ignore these at present.) All the aspects of a
thing are real, whereas the thing is a merely logical
construction. It has, however, the merit of being

THE EXTERNAL WORLD 97

neutral as between different points of view, and of
being visible to more than one person, in the only
sense in which it can ever be visible, namely, in the
sense that each sees one of its aspects.

It will be observed that, while each perspective
contains its own space, there is only one space in which
the perspectives themselves axe the dements. There
axe as many private spaces as there are perspectives ;
there are therefore at least as many as there are per-
cipients, and there may be any number of others which
have a merely material existence and are not seen by
anyone. But there is only one perspective-space,
whose elements are single perspectives, each with
its own private space. We have now to explain
how the private space of a single perspective is cor-
related with part of the one all-embracing perspective
space.

Perspective space is the system of " points of view "
of private spaces (perspectives), or, since " points of
view " have not been defined, we may say it is the
system of the private spaces themselves. These
private spaces will each count as one point, or at any
rate as one element, in perspective space. They are
ordered by means of their similarities. Suppose, for
example, that we start from one which contains the
appearance of a circular disc, such as would be called
a penny, and suppose this appearance, in the perspec-
tive in question, is circular, not elliptic. We can then
f orm a whole series of perspectives containing a gradu-
ated series of circular aspects of varying sizes : for this
purpose we only have to move (as we say) towards
the penny or away from it. The perspectives in which
the penny looks circular will be said to lie on a straight
line in perspective space, and their order on this line
will be that of the sizes of the circular aspects. More-

7

98 SCIENTIFIC METHOD IN PHILOSOPHY

over though, this statement must be noticed and
subsequently examined the perspectives in which the
penny looks big will be said to be nearer to the penny
than those in which it looks small. It is to be remarked
also that any other " thing " than our penny might
have been chosen to define the relations of our per-
spectives in perspective space, and that experience
shows that the same spatial order of perspectives
would have resulted.

In order to explain the correlation of private spaces
with perspective space, we have first to explain what
is meant by " the place (in perspective space) where
a thing is." For this purpose, let us again consider
the penny which appears in many perspectives. We
formed a straight line of perspectives in which the penny
looked circular, and we agreed that those in which it
looked larger were to be considered as nearer to the
penny. We can form another straight line of perspec-
tives in which the penny is seen end-on and looks
like a straight line of a certain thickness. These two
lines will meet in a certain place in perspective space,
i.e. in a certain perspective, which may be defined as
" the place (in perspective space) where the penny is."
It is true that, in order to prolong our lines until they
reach this place, we shall have to make use of other
things besides the penny, because, so far as experience
goes, the penny ceases to present any appearance
after we have come so near to it that it touches the
eye. But this raises no real difficulty, because the
spacial order of perspectives is found empirically
to be independent of the particular " things " chosen
for defining the order. We can, for example, remove
our penny and prolong each of our two straight lines
up to their intersection by placing other pennies
further off in such a way that the aspects of the one are

THE EXTERNAL WORLD 99

circular where those of our original penny were circular,
and the aspects of the other are straight where those
of our original penny were straight. There will then be
just one perspective in which one of the new pennies
looks circular and the other straight. This will be, by
definition, the place where the original penny was in
perspective space.

The above is, of course, only a first rough sketch of
the way in which our definition is to be reached. It
neglects the size of the penny, and it assumes that we
can remove the penny without being disturbed by any
simultaneous changes in the positions of other things.
But it is plain that such niceties cannot affect the
principle, and can only introduce complications in
its application.

Having now defined the perspective, which is the
place where a given thing is, we can understand what
is meant by saying that the perspectives in which
a think looks large are nearer to the things than
those in which it looks small: they are, in fact,
nearer to the perspective which is the place where the
thing is.

We can now also explain the correlation between a
private space and parts of perspective space. If there
is an aspect of a given thing in a certain private space,
then we correlate the place where this aspect is in the
private space with the place where the thing is in
perspective space.

We may define " here " as the place, in perspective
space, which is occupied by our private world. Thus
we can now understand what is meant by speaking of
a thing as near to or far from " here." A thing is near
to " here " if the place where it is is near to my private
world. We can also understand what is meant by saying
that our private world is inside our head ; for our

ioo SCIENTIFIC METHOD IN PHILOSOPHY

private world is a place in perspective space, and may
be part of the place where our head is.

It will be observed that two places in perspective
space are associated with every aspect of a thing:
namely, the place where the thing is, and the place
which is the perspective of which the aspect in question
forms part. Every aspect of a thing is a member
of two different classes of aspects, namely : (i) the
various aspects of the thing, of which at most one
appears in any given perspective ; (2) the perspective
of which the given aspect is a member, i.e. that in which
the thing has the given aspect. The physicist naturally
classifies aspects in the first way, the psychologist in
the second. The two places associated with a single
aspect correspond to the two ways of classifying it.
We may distinguish the two places as that at which,
and that from which, the aspect appears. The " place
at which " is the place of the thing to which the aspect
belongs ; the " place from which " is the place of the
perspective to which the aspect belongs.

Let us now endeavour to state the fact that the aspect
which a thing presents at a given place is affected by
the intervening medium. The aspects of a thing in
different perspectives are to be conceived as spreading
outwards from the place where the thing is, and
undergoing various changes as they get further away
from this place. The laws according to which they
change cannot be stated if we only take account of
the aspects that are near the thing, but require that
we should also take account of the things that are
at the places from which these aspects appear. This
empirical fact can, therefore, be interpreted in terms of
our construction.

We have now constructed a largely hypothetical
picture of the world, which contains and places the

THE EXTERNAL WORLD 101

experienced facts, including those derived from testi-
mony. The world we have constructed can, with a
certain amount of trouble, be used to interpret the
crude facts of sense, the facts of physics, and the facts
of physiology. It is therefore a world which may
be actual. It fits the facts, and there is no empirical
evidence against it ; it also is free from logical im-
possibilities. But have we any good reason to suppose
that it is real ? This brings us back to our original
problem, as to the grounds for believing in the existence
of anything outside my private world. What we have
derived from our hypothetical construction is that there
are no grounds against the truth of this belief, but we
have not derived any positive grounds in its favour.
We will resume this inquiry by taking up again the
question of testimony and the evidence for the existence
of other minds.

It must be conceded to begin with that the argument
in favour of the existence of other people's minds
cannot be conclusive. A phantasm of our dreams will
appear to have a mind a mind to be annoying, as
a rule. It will give unexpected answers, refuse to con-
form to our desires, and show all those other signs
of intelligence to which we are accustomed in the
acquaintances of our waking hours. And yet, when
we are awake, we do not believe that the phantasm
was, like the appearances of people in waking life,
representative of a private world to which we have
no direct access. If we are to believe this of the people
we meet when we are awake, it must be on some ground
short of demonstration, since it is obviously possible
that what we call waking life may be only an unusually
persistent and recurrent nightmare. It may be that
our imagination brings forth all that other people
seem to say to us, all that we read in books, all the

102 SCIENTIFIC METHOD IN PHILOSOPHY

daily, weekly, monthly, and quarterly journals that
distract our thoughts, all the advertisements of soap
and all the speeches of politicians. This may be true,
since it cannot be shown to be false, yet no one can
really believe it. Is there any logical ground for regard-
ing this possibility as improbable ? Or is there nothing
beyond habit and prejudice ?

The minds of other people are among our data, in
the very wide sense in which we used the word at first.
That is to say, when we first begin to reflect, we find
ourselves already believing in them, not because of
any argument, but because the belief is natural to us.
It is, however, a psychologically derivative belief,
since it results from observation of people's bodies ;
and along with other such beliefs, it does not belong
to the hardest of hard data, but becomes, under the
influence of philosophic reflection, just sufficiently
questionable to make us desire some argument con-
necting it with the facts of sense.

The obvious argument is, of course, derived from
analogy. Other people's bodies behave as ours do when
we have certain thoughts and feelings; hence, by
analogy, it is natural to suppose that such behaviour
is connected with thoughts and feelings like our own.
Someone says " Look out 1 " and we find we are on the
point of being killed by a motor-car; we therefore
attribute the words we heard to the person in question
having seen the motor-car first, in which case there are
existing things of which we are not directly conscious.
But this whole scene, with our inference, may occur
in a dream, in which case the inference is generally
considered to be mistaken. Is there anything to make
the argument from analogy more cogent when we are
(as we think) awake ?

The analogy in waking life is only to be preferred to

THE EXTERNAL WORLD 103

that in dreams on the ground of its greater extent and
consistency. If a man were to dream every night about
a set of people whom he never met by day, who had
consistent characters and grew older with the lapse
of years, he might, like the man in Calderon's play,
find it difficult to decide which was the dream-world
and which was the so-called ' * real " world. It is only
the failure of our dreams to form a consistent whole
either with each other or with waking life that makes
us condemn them. Certain uniformities are observed
in waking life, while dreams seem quite erratic. The
natural hypothesis would be that demons and the spirits
of the dead visit us while we sleep ; but the modern
mind, as a rule, refuses to entertain this view, though
it is hard to see what could be said against it. On the
other hand, the mystic, in moments of illumination,
seems to awaken from a sleep which has filled all his
mundane life : the whole world of sense becomes
phantasmal, and he sees, with the clarity and convinc-
ingness that belongs to our morning realization after
dreams, a world utterly different from that of our daily
cares and troubles. Who shall condemn him ? Who
shall justify him ? Or who shall justify the seeming
solidity of the common objects among which we sup-
pose ourselves to live ?

The hypothesis that other people have minds must,
I think, be allowed to be not susceptible of any very
strong support from the analogical argument. At the
same time, it is a hypothesis which systematizes
a vast body of facts and never leads to any consequences
which there is reason to think false. There is therefore
nothing to be said against its truth, and good reason to
use it as a working hypothesis. When once it is
admitted, it enables us to extend our knowledge of
the sensible world by testimony, and thus leads to the

104 SCIENTIFIC METHOD IN PHILOSOPHY

system of private worlds which we assumed in our
hypothetical construction. In actual fact, whatever
we may try to think as philosophers, we cannot help
believing in the minds of other people, so that the
question whether our belief is justified has a merely
speculative interest. And if it is justified, then there is
no further difficulty of principle in that vast extension
of our knowledge, beyond our own private data, which
we find in science and common sense.

This somewhat meagre conclusion must not be
regarded as the whole outcome of our long discussion.
The problem of the connection of sense with objective
reality has commonly been dealt with from a standpoint
which did not carry initial doubt so far as we have
carried it ; most writers, consciously or unconsciously,
have assumed that the testimony of others is to be
admitted, and therefore (at least by implication) that
others have minds. Their difficulties have arisen
after this admission, from the differences in the appear-
ance which one physical object presents to two people
at the same time, or to one person at two times between
which it cannot be supposed to have changed. Such
difficulties have made people doubtful how far objective
reality could be known by sense at all, and have made
them suppose that there were positive arguments
against the view that it can be so known. Our hypo-
thetical construction meets these arguments, and
shows that the account of the world given by common
sense and physical science can be interpreted in a way
which is logically unobjectionable, and finds a place
for all the data, both hard and soft. It is this hypotheti-
cal construction, with its reconciliation of psychology
and physics, which is the chief outcome of our
discussion. Probably the construction is only in
part necessary as an initial, assumption, and can be

THE EXTERNAL WORLD 105

obtained from more slender materials by the logical
methods of which we shall have an example in the
definitions of points, instants, and particles ; but I
do not yet know to what lengths this diminution in
our initial assumptions can be carried.

LECTURE IV

THE WORLD OF PHYSICS AND THE WORLD

OF SENSE

AMONG the objections to the reality of objects of sense,
there is one which is derived from the apparent differ-
ence between matter as it appears in physics and
things as they appear in sensation. Men of science,
for the most part, are willing to condemn immediate
data as " merely subjective," while yet maintaining
the truth of the physics inferred from those data.
But such an attitude, though it may be capable of
justification, obviously stands in need of it ; and the
only justification possible must be one which exhibits
matter as a logical construction from sense-data
unless, indeed, there were some wholly a priori prin-
ciple by which unknown entities could be inferred
from such as are known. It is therefore necessary to
find some way of bridging the gulf between the world
of physics and the world of sense, and it is this problem
which will occupy us in the present lecture. Physicists
appear to be unconscious of the gulf, while psycholo-
gists, who are conscious of it, have not the mathe-
matical knowledge required for spanning it. The
problem is difficult, and I do not know its solution in
detail. All that I can hope to do is to make the
problem felt, and to indicate the kind of methods by
which a solution is to be sought.

WORLDS OF PHYSICS AND OF SENSE 107

Let us begin by a brief description of the two con-
trasted worlds. We will take first the world of physics,
for, though the other world is given while the physical
world is inferred, to us now the world of physics is
the more familiar, the world of pure sense having
become strange and difficult to rediscover. Physics
started from the common-sense belief in fairly per-
manent and fairly- rigid bodiestables and chairs,
stones, mountains, the earth and moon and sun.
This common-sense belief, it should be noticed, is a
piece of audacious metaphysical theorizing; objects
are not continually present to sensation, and it may
be doubted whether they are there when they are not
seen or felt. This problem, which has been acute
since the time of Berkeley, is ignored by common
sense, and has therefore hitherto been ignored by
physicists. We have thus here a first departure from
the immediate data of sensation, though it is a depar-
ture merely by way of extension, and was probably
made by our savage ancestors in some very remote
prehistoric epoch.

But tables and chairs, stones and mountains, are
not quite permanent or quite rigid. Tables and chairs
lose their legs, stones axe split by frost, and mountains
are cleft by earthquakes and eruptions. Then there
are other things, which seem material, and yet present
almost no permanence or rigidity. Breath, smoke,
clouds, are examples of such things so, in a lesser
degree, are ice and snow ; and rivers and seas, though
fairly permanent, are not in any degree rigid. Breath,
smoke, clouds, and generally things that can be seen
but not touched, were thought to be hardly real ; to
this day the usual mark of a ghost is that it can be
seen but not touched. Such objects were peculiar in
the fact that they seemed to disappear completely,

io8 SCIENTIFIC METHOD IN PHILOSOPHY

not merely to be transformed into something else.
Ice and snow, when they disappear, are replaced by
water ; and it required no great theoretical effort to
invent the hypothesis that the water was the same
thing as the ice and snow, but in a new form. Solid
bodies, when they break, break into parts which are
practically the same in shape and size as they were
before. A stone can be hammered into a powder,
but the powder consists of grains which retain the
character they had before the pounding. Thus the
ideal of absolutely rigid and absolutely permanent
bodies, which early physicists pursued throughout
the changing appearances, seemed attainable by
supposing ordinary bodies to be composed of a vast
number of tiny atoms. This billiard-ball view of
matter dominated the imagination of physicists until
quite modern times, until, in fact, it was replaced by
the electromagnetic theory, which in its turn has
developed into a new atomism. Apart from the special
form of the atomic theory which was invented for the
needs of chemistry, some kind of atomism dominated
the whole of traditional dynamics, and was implied in
every statement of its laws and axioms.

The modern form of atomism regards all matter
as composed of two kinds of units, electrons and protons,
both indestructible. All electrons, so far as we can dis-
cover, are exactly alike, and so are all protons. In
addition to this form of atomicity, which is not very
different from that of the Greeks except in being based
upon experimental evidence, there is a wholly new
form, introduced by the theory of quanta. Here the
indivisible unit is a unit of " action," i.e. energy multi-
plied by time, or mass multiplied by length multiplied
by velocity. This is not at all the sort of quantity in
which traditional notions had led us to expect atom-

WORLDS OF PHYSICS AND OF SENSE 109

icity. But relativity makes this kind of atomicity
less surprising, although so fax it cannot deduce any
form of atomicity, either old or new, from its funda-
mental axioms. Relativity has introduced a wholly
novel analysis of physical concepts, and has made it
easier than it formerly was to build a bridge from
physics to sense-data. To make this dear, it will
be necessary to say something about relativity. But
before doing so, let us examine our problem from the
other end, namely that of sense-data.

In the world of immediate data nothing is per-
manent ; even the things that we regard as fairly
permanent, such as mountains, only become data
when we see them, and are not immediately given as
existing at other moments. So far from one all-
embracing space being given, there are several spaces
for each person, according to the different senses which
may be called spatial. Experience teaches us to
obtain one space from these by correlation, and
experience, together with instinctive theorizing, teaches
us to correlate our spaces with those which we believe
to exist in the sensible world of other people. The
construction of a single time offers less difficulty so
long as we confine ourselves to one person's private
world, but the correlation of one private time with
another is a matter of great difficulty. While engaged
in the necessary logical constructions, we can console
ourselves with the knowledge that permanent things,
space, and time have ceased to be, for relativity
physics, part of the bare bones of the world, and are
now admitted to be constructions. In attempting to
construct them from sense-data and particulars struc-
turally analogous to sense-data, we are, therefore,
only pushing the procedure of relativity theory one
stage further back.

no SCIENTIFIC METHOD IN PHILOSOPHY

The belief in indestructible " things " very early
took the form of atomism. The underlying motive in
atomism was not, I think, any empirical success in
interpreting phenomena, but rather an instinctive
belief that beneath all the changes of the sensible
world there must be something permanent and un-
changing. This belief was, no doubt, fostered and
nourished by its practical successes, culminating in
the conservation of mass ; but it was not produced
by these successes. On the contrary, they were
produced by it. Philosophical writers on physics
sometimes speak as though the conservation of some-
thing or other were essential to the possibility of
science, but this, I believe, is an entirely erroneous
opinion. If the a priori belief in permanence had
not existed, the same laws which are now formu-
lated in terms of this belief might just as well
have been formulated without it. Why should we
suppose that, when ice melts, the water which replaces
it is the same thing in a new form ? Merely because
this supposition enables us to state the phenomena
in a way which is consonant with our prejudices.
What we really know is that, under certain conditions
of temperature, the appearance we call ice is replaced
by the appearance we call water. We can give laws
according to which the one appearance will be succeeded
by the other, but there is no reason except prejudice
for regarding both as appearances of the same
substance.

One task, if what has just been said is correct,
which, confronts us in trying to connect the world of
sense with the world of physics, is the task of recon-
structing the conception of matter without the a
priori beliefs which historically gave rise to it. In
spite of the revolutionary results of modern physics,

WORLDS OF PHYSICS AND OF SENSE in

the empirical successes of the conception of matter
show that there must be some legitimate conception
which fulfils roughly the same functions. The time
has hardly come when we can state precisely what
this legitimate conception is, but we can see in a
general way what it must be like. For this purpose,
it is only necessary to take our ordinary common-sense
statements and reword them without the assumption
of permanent substance. We say, for example, that
things change gradually sometimes very quickly,
but not without passing through a continuous series
of intermediate states, or at least an approximately
continuous series, if the discontinuities of the quantum
theory should prove -ultimate. What this means is
that, given any sensible appearance, there will usually
be, if we watch, a continuous series of appearances
connected with the given one, leading on by imper-
ceptible gradations to the new appearances which
common sense regards as those of the same thing.
Thus a thing may be defined as a certain series of
appearances, connected with each other by continuity
and by certain causal laws. In the case of slowly
changing things, this is easily seen. Consider, say, a
wall-paper which fades in the course of years. It is
an effort not to conceive of it as one " thing " whose
colour is slightly different at one time from what it is
at another. But what do we really know about it ?
We know that under suitable circumstances^-i.e. when
we are, as is said, " in the room "we perceive certain
colours in a certain pattern : not always precisely the
same colours, but sufficiently similar to fed familiar.
If we can state the laws according to which the colour
varies, we can state all that is empirically verifiable ;
the assumption that there is a constant entity, the
wall-paper, which "has" these various colours at

H2 SCIENTIFIC METHOD IN PHILOSOPHY

various times, is a piece of gratuitous metaphysics.
We may, if we like, define the wall-paper as the series
of its aspects. These are collected together by the
same motives which led us to regard the wall-paper
as one thing, namely a combination of sensible con-
tinuity and causal connection. More generally, a
" thing " will be defined as a certain series of aspects,
namely those which would commonly be said to be
o/the thing. To say that a certain aspect is an aspect
of a certain thing will merely mean that it is one of
those which, taken serially, are the thing. Everything
will then proceed as before : whatever was verifiable
is unchanged, but our language is so interpreted as to
avoid an unnecessary metaphysical assumption of
permanence.

The above extrusion of permanent things affords an
example of the maxim which inspires all scientific
philosophizing, namely " Occam's razor " : Entities are
not to be multiplied without necessity. In other
words, in dealing with any subject-matter, find out
what entities are undeniably involved, and state
everything in terms of these entities. Very often the
resulting statement is more complicated and difficult
than one which, like common sense and most philo-
sophy, assumes hypothetical entities whose existence
there is no good reason to believe in. We find it easier
to imagine a wall-paper with changing colours than to
think merely of the series of colours ; but it is a mistake
to suppose that what is easy and natural in thought is
what is most free from unwarrantable assumptions, as
the case of " things " very aptly illustrates.

The above summary account of the genesis of
"things," though it may be correct in outline, has
omitted some serious difficulties which it is necessary
briefly to consider. Starting from a world of helter-

WORLDS OF PHYSICS AND OF SENSE 113

skelter sense-data, we wish to collect them into series,
each of which can be regarded as consisting of the
successive appearances of one " thing." There is, to
begin with, some conflict between what common sense
regards as one thing, and what physics regards an
unchanging collection of particles. To common sense,
a human body is one thing, but to science the matter
composing it is continually changing. This conflict,
however, is not very serious, and may, for our rough
preliminary purpose, be largely ignored. The problem
is : by what principles shall we select certain data
from the chaos, and call them all appearances of the
same thing ?

A rough and approximate answer to this question
is not very difficult. There are certain fairly stable
collections of appearances, such as landscapes, the
furniture of rooms, the faces of acquaintances. In
these cases, we have little hesitation in regarding them
on successive occasions as appearances of one thing or
collection of things. But, as the Comedy of Errors
illustrates, we may be led astray if we judge by mere
resemblance. This shows that something more is
involved, for two difierent things may have any degree
of likeness up to exact similarity.

Another insufficient criterion of one thing is con-
tinuity. As we have already seen, if we watch what
we regard as one changing thing, we usually find its
changes to be continuous so fax as our senses can
perceive. We are thus led to assume that, if we see
two finitely different appearances at two different times,
and if we have reason to regard them as belonging
to the same thing, then there was a continuous series
of intermediate states of that thing during the time
when we were not observing it. And so it comes to be
thought that continuity of change is necessary and

8

114 SCIENTIFIC METHOD IN PHILOSOPHY

sufficient to constitute one thing. But in fact it is
neither. It is not necessary, because the unobserved
states, in the case where our attention has not been
concentrated on the thing throughout, are purely
hypothetical, and cannot possibly be our ground for
supposing the earlier and later appearances to belong
to the same thing ; on the contrary, it is because we
suppose this that we assume intermediate unobserved
states. Continuity is also not sufficient, since we can,
for example, pass by sensibly continuous gradations
from any one drop of the sea to any other drop. The
utmost we can say is that discontinuity during un-
interrupted observation is as a rule a mark of difference
between things, though even this cannot be said in
such cases as sudden explosions. (We are speaking
throughout of the immediate sensible appearance,
counting as continuous whatever seems continuous,
and as discontinuous whatever seems discontinuous.)
The assumption of continuity is, however, success-
fully made in physics. This proves something, though
not anything of very obvious utility to our present
problem : it proves that nothing in the known world
(apart, possibly, from quantum phenomena) is incon-
sistent with the hypothesis that all changes are really
continuous, though from too great rapidity or from
our lack of observation they may not always appear
continuous. In this hypothetical sense, continuity or
change which, though sudden, is in accordance with
quantum principles, may be allowed to be a necessary
conidtion if two appearances are to be classed as
appearances of the same thing. But it is not a sufficient
condition, as appears from the instances of the drops
in the sea. Thus something more must be sought
before we can give even the roughest definition of a
" thing."

WORLDS OP PHYSICS AND OF SENSE 115

What is wanted further seems to be something in
the nature of fulfilment of causal laws. This statement
as it stands, is very vague, but we will endeavour to
give it precision. When I speak of " causal laws," I
mean any laws which connect events at different times,
or even, as a limiting case, events at the same time
provided the connection is not logically demonstrable.
In this very general sense, the laws of dynamics are
causal laws, and so are the laws correlating the simul-
taneous appearances of one " thing " to different
senses. The question is : How do such laws help in
the definition of a " thing " ?

To answer this question, we must consider what it
is that is proved by the empirical success of physics.
What is proved is that its hypotheses, though un-
verifiable where they go beyond sense-data, are at no
point in contradiction with sense-data, but, on the
contrary, are ideally such as to render all sense-data
calculable from a sufficient collection of data all belong-
ing to a given period of time. Now physics has found
it empirically possible to collect sense-data into series,
each series being regarded as belonging to one " thing,"
and behaving, with regard to the laws of physics, in
a way in which series not belonging to one thing would
in general not behave. If it is to be unambiguous
whether two appearances belong to the same thing or
not, there must be only one way of grouping appear-
ances so that the resulting things obey the laws of
physics. It would be very difficult to prove that this
is the case, but for our present purposes we may let
this point pass, and assume that there is only one
way. We must include in our definition of a " thing "
those of its aspects, if any, which are not observed.
Thus we may lay down the following definition :
Things are those series of aspects which obey the laws of

n6 SCIENTIFIC METHOD IN PHILOSOPHY

physics. That such series exist is an empirical fact,
which constitutes the verifiability of physics.

It may still be objected that the "matter" of
physics is something other than series of sense-data.
Sense-data, it may be said, belong to psychology and
are, at any rate in some sense, subjective, whereas
physics is quite independent of psychological con-
siderations, and does not assume that its matter only
exists when it is perceived.

To this objection there are two answers, both of
some importance.

(a) We have been considering, in the above account,
the question of the verifiability of physics. Now
verifiability is by no means the same thing as truth ;
it is, in fact, something far more subjective and
psychological. For a proposition to be verifiable, it
is not enough that it should be true, but it must also
be such as we can discover to be true. Thus verifiability
depends upon our capacity for acquiring knowledge,
and not only upon the objective truth. In physics,
as ordinarily set forth, there is much that is unverifi-
able : there are hypotheses as to (a) how things would
appear to a spectator in a place where, as it happens,
there is no spectator ; (j3) how things would appear
at times when, in fact, they are not appearing to
anyone ; (y) things which never appear at all. All
these are introduced to simplify the statement of
the causal laws, but none of them form an integral
part of what is known to be true in physics. This
brings us to our second answer.

(b) If physics is to consist wholly of propositions
known to be true, or at least capable of being proved
or disproved, the three kinds of hypothetical entities
we have just enumerated must all be capable of being
exhibited as logical functions of sense-data. In order

WORLDS OF PHYSICS AND OF SENSE 117

to show how this might possibly be done, let us recall
the hypothetical Ldbnizian universe of Lecture III.
In that universe, we had a number of perspectives,
two of which never had any entity in common, but
often contained entities which could be sufficiently
correlated to be regarded as belonging to the same
thing. We will call one of these an " actual " private
world when there is an actual spectator to which it
appears, and " ideal " when it is merely constructed
on principles of continuity. A physical thing consists,
at each instant, of the whole set of its aspects at that
instant, in all the different worlds ; thus a momentary
state of a thing is a whole set of aspects. An " ideal "
appearance will be an aspect merely calculated, but
not actually perceived by any spectator. An " ideal "
state of a thing will be a state at a moment when all
its appearances are ideal. An ideal thing will be one
whose states at all times are ideal. Ideal appearances,
states, and things, since they are calculated, must be
functions of actual appearances, states, and things ;
in fact, ultimately, they must be functions of actual
appearances. Thus it is unnecessary, for the enuncia-
tion of the laws of physics, to assign any reality to
ideal elements : it is enough to accept them as logical
constructions, provided we have means of knowing
how to determine when they become actual. This,
in fact, we have with some degree of approximation ;
the starry heaven, for instance, becomes actual when-
ever we choose to look at it. It is open to us to believe
that the ideal dements exist, and there can be no
reason for disbelieving this ; but unless in virtue of
some a priori law we cannot know it, for empirical
knowledge is confined to what we actually observe.

We come now to the conception of space. Here it
is of the greatest importance to distinguish sharply

n8 SCIENTIFIC METHOD IN PHILOSOPHY

between the space of physics and the space of one
man's experience. It is the latter that must concern
us first.

People who have never read any psychology seldom
realize how much mental labour has gone into the
construction of the one all-embracing space into which
all sensible objects are supposed to fit. Kant, who
was unusually ignorant of psychology, described space
as " an infinite given whole," whereas a moment's
psychological reflection shows that a space which is
infinite is not given, while a space which can be called
given is not infinite. What the nature of " given "
space really is, is a difficult question, upon which
psychologists are by no means agreed. But some
general remarks may be made, which will suffice to
show the problems, without taking sides on any
psychological issue still in debate.

The first thing to notice is that different senses have
different spaces. The space of sight is quite different
from the space of touch : it is only by experience in
infancy that we learn to correlate them. In later life,
when we see an object within reach, we know how to
touch it, and more or less what it will fed like ; if we
touch an object with our eyes shut, we know where we
should have to look for it, and more or less what it
would look like. But this knowledge is derived from
early experience of the correlation of certain kinds of
touch-sensations with certain kinds of sight-sensations.
The one space into which both kinds of sensations fit
is an intellectual construction, not a datum. And
besides touch and sight, there are other kinds of
sensation which give other, though less important
spaces : these also have to be fitted into the one space
by means of experienced correlations. And as in the
case of things, so here : the one all-embracing space,

WORLDS OF PHYSICS AND OF SENSE 119

though convenient as a way of speaking, need not be
supposed really to exist. All that experience makes
certain is the several spaces of the several senses
correlated by empirically discovered laws. . The one
space may turn out to be valid as a logical construction,
compounded of the several spaces, but there is no good
reason to assume its independent metaphysical reality.

Another respect in which the spaces of immediate
experience differ from the space of geometry and
physics is in regard to points, The space of geometry
and physics consists of an infinite number of points, but
no one has ever seen or touched a point. If there are
points in a sensible space, they must be an inference.
It is not easy to see any way in which, as independent
entities, they could be validly inferred from the data ;
thus here again, we shall have, if possible, to find
some logical construction, some complex assemblage
of immediately given objects, which will have the
geometrical properties required of points. It is cus-
tomary to think of points as simple and infinitely small,
but geometry in no way demands that we should flwilr
of them in this way. All that is necessary for geometry
is that they should have mutual relations possessing
certain enumerated abstract properties, and it may be
that an assemblage of data of sensation will serve this
purpose. Exactly how this is to be done I do not
yet know, but it seems fairly certain that it can be
done.

An illustrative method, simplified so as to be easily
manipulated, has been invented by Dr. Whitehead
for the purpose of showing how points might be manu-
factured from sense-data together with other structur-
ally analogous particulars. This method is set forth
in his Principles of Natural Knowledge (Cambridge,
1919) and Concept of Nature (Cambridge, 1920). It

120 SCIENTIFIC METHOD IN PHILOSOPHY

is impossible to explain this method more concisely
than in those books, to which the reader is therefore
referred. But a few words may be said by way of
explaining the general principles underlying the method.
We have first of all to observe that there are no infini-
tesimal sense-data : any surface we can see, for example,
must be of some finite extent. We assume that this
applies, not only to sense-data, but to the whole of
the stuff composing the world : whatever is not an
abstraction has some finite spatio-temporal size,
though we cannot discover a lower limit to the sizes
that are possible. But what appears as one undivided
whole is often found, under the influence of attention,
to split up into parts contained within the whole.
Thus one spatial datum may be contained within
another, and entirely enclosed by the other. This
relation of enclosure, by the help of some very natural
hypotheses, will enable us to define a " point " as a
certain set of spatial objects ; roughly speaking, the
set will consist of aJl volumes which would naturally
be said to contain the point.

It should be observed that Dr. Whitehead's abstract
logical methods are applicable equally to psychological
space, physical space, time, and space-time. But
as applied to psychological space, they do not yield
continuity unless we assume that sense-data always
contain parts which are not sense-data. Sense-data
have a minimum size, below which nothing is experi-
enced ; but Dr. Whitehead's methods postulate that
there shall be no such minimum. We cannot therefore
construct a continuum without assuming the existence
of particulars which are not experienced. This,
however, does not constitute a real difficulty, since
there is no reason to suppose that the space of our
immediate experience possesses mathematical con-

WORLDS OF PHYSICS AND OF SENSE 121

tinuity. The full employment of Dr. Whitehead's
methods, therefore, belongs rather to physical space
than to the space of experience. This question will
concern us again later, when we come to consider
physical space-time and its partial correlation with the
space and time of experience.

A very interesting attempt to show the kinds of
geometry that can be constructed out of the actual
materials supplied in sensation will be found in Jean
Nicod's La gfom&rie dans le monde sensible (Paris,

1923)-

The question of time, so long as we confine ourselves
to one private world, is rather less complicated than
that of space, and we can see pretty dearly how it
might be dealt with by such methods as we have been
considering. Events of which we are conscious do not
last merely for a mathematical instant, but always for
some finite time, however short. Even if there be a
physical world such as the mathematical theory of
motion supposes, impressions on our sense-organs
produce sensations which are not merely and strictly
instantaneous, and therefore the objects of sense of
which we are immediately conscious are not strictly
instantaneous. Instants, therefore, are not among
the data of experience, and, if legitimate, must be
either inferred or constructed. It is difficult to see
how they can be validly inferred ; thus we are left
with the alternative that they must be constructed.
How is this to be done ?

Immediate experience provides us with two time-
relations among events : they may be simultaneous,
or one may be earlier and the other later. These two
are both part of the crude data ; it is not the case that
only the events are given, and their time-order is added
by our subjective activity. The time-order, within

122 SCIENTIFIC METHOD IN PHILOSOPHY

certain limits, is as much given as the events. In any
story of adventure you will find such passages as the
following : " With a cynical smile he pointed the
revolver at the breast of the dauntless youth. ' At the
word three I shall fire,' he said. The words one and
two had already been spoken with a cool and deliberate
distinctness. The word three was forming on his
lips, M this moment a blinding flash of lightning
rent the air." Here we have simultaneity not due,
as Kant would have us believe, to the subjective
mental apparatus of the dauntless youth, but given as
objectively as the revolver and the lightning. And
it is equally given in immediate experience that the
words one and two come earlier than the flash. These
time-relations hold between events which are not
strictly instantaneous. Thus one event may begin
sooner than another, and therefore be before it, but
may continue after the other has begun, and therefore
be also simultaneous with it. If it persists after the
other is over, it will also be later than the other.
Earlier, simultaneous, and later, are not inconsistent
with each other when we are concerned with events
which last for a finite time, however short ; they
only become inconsistent when we are dealing with
something instantaneous.

It is to be observed that we cannot give what may
be called absolute dates, but only dates determined by
events. We cannot point to a time itseUE, but only
to some event occurring at that time. There is
therefore no reason in experience to suppose that there
are times as opposed to events : the events, ordered
by the relations of simultaneity and succession, axe
all that experience provides. Hence, unless we are
to introduce superfluous metaphysical entities, we
must, in defining what we can regard as an instant,

WORLDS OF PHYSICS AND OF SENSE 123

proceed by means of some construction which assumes
nothing beyond events and their temporal relations.

If we wish to assign a date exactly by means of events,
how shall we proceed ? If we take any one event, we
cannot assign our date exactly, because the event is
not instantaneous, that is to say, it may be simultane-
ous with two events which are not simultaneous with
each other. In order to assign a date exactly, we must
be able, theoretically, to determine whether any given
event is before, at, or after this date, and we must
know that any other date is either before or after this
date, but not simultaneous with it. Suppose, now,
instead of taking one event A, we take two events A
and B, and suppose A and B partly overlap, but B
ends before A ends. Then an event which is simul-
taneous with both A and B must exist during the time
when A and B overlap ; thus we have come rather
nearer to a precise date than when we considered
A and B alone. Let C be an event which is simul-
taneous with both A and B, but which ends before
either A or B has ended. Then an event which is
simultaneous with A and B and C must exist during

the time when all three overlap, which is a still
shorter time. Proceeding in this way, by taking more
and more events, a new event which is dated as
simultaneous with all of them becomes gradually
more and more accurately dated. This suggests a

124 SCIENTIFIC METHOD IN PHILOSOPHY

way by which a completely accurate date can be
defined.

Let us take a group of events of which any two
overlap, so that there is some time, however short,
when they all exist. If there is any other event which
is simultaneous with all of these, let us add it to the
group ; let us go on until we have constructed a
group such that no event outside the group is simul-
taneous with all of them, but all the events inside the
group are simultaneous with each other. Let us
define this whole group as an instant of time. It
remains to show that it has the properties we expect
of an instant.

What are the properties we expect of instants ?
First, they must form a series : of any two, one must
be before the other, and the other must be not before
the one ; if one is before another, and the other before
a third, the first must be before the third. Secondly,
every event must be at a certain number of instants ;
two events are simultaneous if they are at the same
instant, and one is before the other if there is an instant,
at which the one is, which is earlier than some instant
at which the other is. Thirdly, if we assume that
there is always some change going on somewhere
during the time when any given event persists, the
series of instants ought to be compact, i.e. given any
two instants, there ought to be other instants between
them. Do instants, as we have defined them, have
these properties ?

We shall say that an event is " at " an instant when
it is a member of the group by which the instant is
constituted; and we shall say that one instant is
before another if the group which is the one instant
contains an event which is earlier than, but not simul-
taneous with, some event in the group which is the

WORLDS OF PHYSICS AND OF SENSE 125

other instant. When one event is earlier than, but
not simultaneous with another, we shall say that it
" wholly precedes " the other. Now we know that
of two events which belong to one experience but are
not simultaneous, there must be one which wholly
precedes the other, and in that case the other cannot
also wholly precede the one ; we also know that, if
one event wholly precedes another, and the other
wholly precedes a third, then the first wholly pre-
cedes the third. From these facts it is easy to deduce
that the instants as we have defined them form
a series.

We have next to show that every event is " at "
least one instant, i.e. that, given any event, there is
at least one class, such as we used in defining instants,
of which it is a member. For this purpose, consider
all the events which are simultaneous with a given
event, and do not begin later, Le. are not wholly
after anything simultaneous with it. We will call
these the " initial contemporaries of the given event.
It will be found that this class of events is the first
instant at which the given event exists, provided
every event wholly after some contemporary of the
given event is wholly after some initial contemporary
of it.

Finally, the series of instants will be compact if,
given any two events of which one wholly precedes
the other, there are events wholly after the one and
simultaneous with something wholly before the other.
Whether this is the case or not, is an empirical question ;
but if it is not, there is no reason to expect the time-
series to be compact. 1

1 The assumptions made concerning time-relations in one
experience in the above axe as follows :

I. In order to secure that instants form a series, we assume :

124 SCIENTIFIC METHOD IN PHILOSOPHY

way by which a completely accurate date can be
defined.

Let us take a group of events of which any two
overlap, so that there is some time, however short,
when they all exist. If there is any other event which
is simultaneous with all of these, let us add it to the
group ; let us go on until we have constructed a
group such that no event outside the group is simul-
taneous with all of them, but all the events inside the
group are simultaneous with each other. Let us
define this whole group as an instant of time. It
remains to show that it has the properties we expect
of an instant.

What are the properties we expect of instants?
First, they must form a series : of any two, one must
be before the other, and the other must be not before
the one ; if one is before another, and the other before
a third, the first must be before the third. Secondly,
every event must be at a certain number of instants ;
two events are simultaneous if they are at the same
instant, and one is before the other if there is an instant,
at which the one is, which is earlier than some instant
at which the other is. Thirdly, if we assume that
there is always some change going on somewhere
during the time when any given event persists, the
series of instants ought to be compact, i.e. given any
two instants, there ought to be other instants between
them. Do instants, as we have defined them, have
these properties ?

We shall say that an event is " at " an instant when
it is a member of the group by which the instant is
constituted; and we shall say that one instant is
before another if the group which is the one instant
contains an event which is earlier than, but not simul-
taneous with, some event in the group which is the

WORLDS OF PHYSICS AND OF SENSE 125

other instant. When one event is earlier than, but
not simultaneous with another, we shall say that it
" wholly precedes " the other. Now we know that
of two events which belong to one experience but are
not simultaneous, there must be one which wholly
precedes the other, and in that case the other cannot
also wholly precede the one ; we also know that, if
one event wholly precedes another, and the other
wholly precedes a third, then the first wholly pre-
cedes the third. From these facts it is easy to deduce
that the instants as we have defined them form
a series.

We have next to show that every event is "at"
least one instant, i.e. that, given any event, there is
at least one class, such as we used in defining instants,
of which it is a member. For this purpose, consider
all the events which are simultaneous with a given
event, and do not begin later, i.e. are not wholly
after anything simultaneous with it. We will call
these the " imtia.l contemporaries of the given event.
It will be found that this class of events is the first
instant at which the given event exists, provided
every event wholly after some contemporary of the
given event is wholly after some initial contemporary
of it.

Finally, the series of instants will be compact if,
given any two events of which one wholly precedes
the other, there are events wholly after the one and
simultaneous with something wholly before the other.
Whether this is the case or not, is an empirical question ;
but if it is not, there is no reason to expect the time-
series to be compact. 1

i The assumptions made concerning tune-relations in one
experience in the above are as follows :

I. In older to secure that instants form a series, we assume :

126 SCIENTIFIC METHOD IN PHILOSOPHY

Thus our definition of instants secures all that
mathematics requires, without having to assume the
existence of any disputable metaphysical entities.

With regard to compactness in the time of one
experience, there are the same observations to make
as in the case of space. The events which we experi-
ence have not only a finite duration, but a duration
which cannot sink below a certain TniniTnnni ; there-
fore they will only fit into a compact series if we either
bring in events wholly outside our experience, or
assume that experienced events have parts which we
do not experience, or postulate that we can experi-

(*) No event wholly precedes itself. (An " event " is
defined as whatever is simultaneous with some-
thing or other.)

(6) If one event wholly precedes another, and the other
wholly precedes a third, then the first wholly
precedes the third.

(e) If one event wholly precedes another, it is not
simultaneous with it.

(d) Of two events which are not simultaneous, one

must wholly precede the other.

II. In order to secure that the initial contemporaries of a
given event should form an instant, we assume :

(e ) An event wholly after some contemporary of a given

event is wholly after some initial contemporary

of the given event.
III. In order to secure that the series of instants shall be

compact, we assume :
(/) If one event wholly precedes another, there is an

event wholly after the one and simultaneous with

something wholly before the other.

This assumption entails the consequence that if one event
covers the whole of a stretch of time immediately preceding
another event, then it must have at least one instant in common
with the other event ; i.e. it is impossible for one event to cease
just before another begins. I do not know whether this should
be regarded as inadmissible. For a mathematico-logical
treatment of the above topics, cf . K. Wiener, " A Contribution
to the Theory of Relative Position," Proc. Camb. Phil. Soc.,
xvii. 5, pp. 441-449.

WORLDS OF PHYSICS AND OF SENSE 127

ence an Infinite number of events at once. Here,
again, the full application of our logico-mathematical
method is only possible when we come to physical time.
This topic will be discussed again near the end of
Lecture V.

Instants may also be defined by means of the
enclosure-relation, exactly as was done in the case of
points. One object will be temporally enclosed by
another when it is simultaneous with the other, but
not before or after it. Whatever encloses temporally
or is enclosed temporally we shall call an " event."
In order that the relation of temporal enclosure may
lead to instants we require (i) that it should be tran-
sitive, i.e. that if one event encloses another, and
the other a third, then the first encloses the third ;
(2) that every event encloses itself, but if one event
encloses another different event, then the other does
not enclose the one ; (3) that given any set of events
such that there is at least one event enclosed by all
of them, then there is an event enclosing all that they
all enclose, and itself enclosed by all of them ; (4) that
there is at least one event. To ensure infinite divisi-
bility, we require also that every event should enclose
events other than itself. Assuming these character-
istics, temporal enclosure can be made to give rise to a
compact series of instants. We can now form an " en- .
closure-series " of events, by choosing a group of events
such that of any two there is one which encloses the
other ; this will be a " punctual enclosure-series " if,
given any other enclosure-series such that every member
of our first series encloses some member of our second,
then every member of our second series encloses some
member of our first. Then an " instant " is the class
of all events which enclose members of a given punctual
enclosure-series.

128 SCIENTIFIC METHOD IN PHILOSOPHY

The correlation of the times of different private
worlds is a more difficult matter. We saw, in Lecture
III, that different private worlds often contain corre-
lated appearances, such as common sense would
regard as appearances of the same "thing." When
two appearances in different worlds are so correlated
as to belong to one momentary " state " of a thing,
it would be natural to regard them as simultaneous,
and as thus affording a simple means of correlating
different private times. But this can only be regarded
as a first approximation. What we call one sound
will be heard sooner by people near the source of the
sound than by people further from it, and the same
applies, though in a less degree, to light. Thus two
correlated appearances in different worlds are not
necessarily to be regarded as occurring at the same
date in physical time, though they will be parts of one
momentary state of a thing. The correlation of
different private times is regulated by the desire to
secure the simplest possible statement of the laws of
physics, and thus raises rather complicated technical
problems; these problems are dealt with by the
theory of relativity, and show that it is impossible
validly to construct one all-embracing time having
any physical significance.

The above brief outline, must not be regarded as
more than tentative and suggestive. It is intended
merely to show the kind of way in which, given a
world with the kind of properties that psychologists
find in the world of sense, it may be possible, by
means of purely logical constructions, to make it
amenable to mathematical treatment by defining
series or classes of sense-data which can be called
respectively particles, points, and instants. If such
constructions are possible, then mathematical physics

WORLDS OF PHYSICS AND OF SENSE 129

is applicable to the real world, in spite of the fact that
its particles, points, and instants are not to be found
among actually existing entities.

The space-time of physics has not a very dose
relation to the space and time of the world of one
person's experience. Everything that occurs in one
person's experience must, from the standpoint of
physics, be located within that person's body ; this
is evident from considerations of causal continuity.
What occurs when I see a star occurs as the result of
light-waves impinging on the retina, and causing a
process in the optic nerve and brain ; therefore the
occurrence called "seeing a star" must be in the
brain. If we define a piece of matter as a set of events
(as was suggested above), the sensation of seeing a
star will be one of the events which are the brain of
the percipient at the time of the perception. Thus
every event that I experience will be one of the events
which constitute some part of my body. The space
of (say) my visual perceptions is only correlated, with
physical space, more or less approximately ; from the
physical point of view, whatever I see is inside my
head. I do not see physical objects; I see effects
which they produce in the region where my brain is.
The correlation of visual and physical space is rendered
approximate by the fact that my visual sensations are
not wholly due each to some physical object, but also
partly to the intervening medium. Further, the rela-
tion of visual sensation to physical object is one-many,
not not-one, because our senses are more or less vague :
things which look different under the microscope may
be indistinguishable to the naked eye. The inferences
from perceptions to physical facts depend always upon
causal laws, which enable us to bring past history to
bear ; e.g. if we have just examined an object under a

Q

130 SCIENTIFIC METHOD IN PHILOSOPHY

microscope, we assume that it is still very similar to
what we then saw it to be, or rather, to what we
inferred it to be from what we then saw. It is through
history and testimony, together with causal laws,
that we arrive at physical knowledge which is much
more precise than anything inferable from the percep-
tions of one moment. History, testimony, and causal
laws axe, of course, in their various degrees, open to
question. But we are not now considering whether
physics is true, but how, if it is true, its world is related
to that of the senses.

With regard to time, the relation of psychology to
physics is surprisingly simple. The time of our
experience is the time which results, in physics, from
taking our own body as the origin. Seeing that
all the events in my experience are, for physics, in
my body, the time-interval between them is what
relativity theory calls the " interval " (in space-time)
between them. Thus the time-interval between two
events in one person's experience retains a direct
physical significance in the theory of relativity. But
the merging of physical space and time into space-
time does not correspond to anything in psychology.
Two events which are simultaneous in my experience
may be spatially separate in psychical space, e.g.
when I see two stars at once. But in physical space
these two events are not separated, and indeed they
occur in the same place in space-time. Thus in this
respect relativity theory has complicated the relation
between perception and physics.

The problem which the above considerations are
intended to elucidate is one whose importance and
even existence has been concealed by the unfortunate
separation of different studies which prevails through-
out the civilized world. Physicists, ignorant and con-

WORLDS OF PHYSICS AND OF SENSE 131

temptuous of philosophy, have been content to assume
.their particles, points, and instants in practice, while
conceding, with ironical politeness, that their concepts
laid no claim to metaphysical validity. Metaphysi-
cians, obsessed by the idealistic opinion that only mind
is real, and the Parmenidean belief that the real is
unchanging, repeated one after another the supposed
contradictions in the notions of matter, space, and
time, and therefore naturally made no endeavour to
invent a tenable theory of particles, points, and
instants. Psychologists, who have done invaluable
work in bringing to light the chaotic nature of the
crude materials supplied by unmanipulated sensation,
have been ignorant of mathematics and modern
logic, and have therefoie been content to say that
matter, space, and time are "intellectual construc-
tions," without making any attempt to show in detail
either how the intellect can construct them, or what
secures the practical validity which physics shows
them to possess. Philosophers, it is to be hoped, will
come to recognize that they cannot achieve any solid
success in such problems without some slight knowledge
of logic, mathematics, and physics ; meanwhile, for
want of students with the necessary equipment, this
vital problem remains unattempted and unknown. 1

There are, it is true, two authors, both physicists,
whotave done something, though not much, to bring
about a recognition of the problem as one demanding
study. These two authors are Poincarfi and Mach,
Poincar6 especially in his Science and Hypothesis,

* This was written in 1914. Since then, largely as a result
of the general theory of relativity, a great deal of valuable
work has been done; I should wish to mention specially
Professor Eddington, Dr. Whitehead, and Dr. Broad, as having
contributed, from different angles, to the solution of the
problems dealt with in this lecture.

132 SCIENTIFIC METHOD IN PHILOSOPHY

Mach especially in his Analysis of Sensations. Both
of them, however, admirable as their work is, seem to
me to suffer from a general philosophical bias. Poin-
car6 is Kantian, while Mach is ultra-empiricist ; with
Poincar6 almost all the mathematical part of physics
is merely conventional, while with Mach the sensation
as a mental event is identified with its object as a
part of the physical world. Nevertheless, both these
authors, and especially Mach, deserve mention as
having made serious contributions to the consideration
of our problem.

When a point or an instant is defined as a class of
sensible qualities, the first impression produced is
likely to be one of wild and wilful paradox. Certain
considerations apply here, however, which will again be
relevant when we come to the definition of numbers.
There is a whole type of problems which can be solved
by such definitions, and almost always there will be
at first an effect of paradox. Given a set of objects
any two of which have a relation of the sort called
"symmetrical and transitive," it is almost certain
that we shall come to regard them as all having some
common quality, or as all having the same relation
to some one object outside the set. This kind of case
is important, and I shall therefore try to make it
clear even at the cost of some repetition of previous
definitions.

A relation is said to be " symmetrical " when, if one
term has this relation to another, then the other also
has it to the one. Thus "brother or sister" is a
1 ' symmetrical " relation : if one person is a brother or
a sister of another, then the other is a brother or
sister of the one. Simultaneity, again, is a symmetrical
relation ; so is equality in size. A relation is said to
be " transitive " when, if one term has this relation to

WORLDS OF PHYSICS AND OF SENSE 133

another, and the other to a third, then the one has it
to the third. The symmetrical relations mentioned
just now are also transitive provided, in the case of
" brother or sister," we allow a person to be counted
as his or her own brother or sister, and provided, in
the case of simultaneity, we mean complete simul-
taneity, i.e. beginning and ending together.

But many relations are transitive without being
symmetrical for instance, such relations as " greater,"
" earlier," "to the right of," " ancestor of," in fact
all such relations as give rise to series. Other relations
are symmetrical without being transitive for example,
difference in any respect. If A is of a different age
from B, and B of a different age from C, it does not
f ollow that A is of a different age from C. Simultaneity,
again, in the case of events which last for a finite time,
will not necessarily be transitive if it only means that
the times of the two events overlap. If A ends just
after B has begun, and B ends just after C has begun,
A and B will be simultaneous in this sense, and so will
B and C, but A and C may well not be simultaneous*

All the relations which can naturally be represented
as equality in any respect, or as possession of a common
property, are transitive and symmetrical this applies,
for example, to such relations as being of the same
height or weight or colour. Owing to the fact that
possession of a common property gives rise to a transi-
tive symmetrical relation, we come to imagine that
wherever such a relation occurs it must be due to a
common property. " Being equally numerous " is a
transitive symmetrical relation of two collections;
hence we imagine that both have a common property,
called their number. " Existing at a given instant "
(in the sense in which we defined an instant) is a
transitive symmetrical relation; hence we come to

I 3 4 SCIENTIFIC METHOD IN PHILOSOPHY

think that there really is an instant which confers a
common property on all the things existing at that
instant. " Being states of a given thing " is a transi-
tive symmetrical relation ; hence we come to imagine
that there really is a thing, other than the series of
states, which accounts for the transitive symmetrical
relation. In all such cases, the class of terms that
have the given transitive symmetrical relation to a
given term will fulfil all the formal requisites of a
common property of all the members of the class.
Since there certainly is the class, while any other
common property may be illusory, it is prudent, in
order to avoid needless assumptions, to substitute the
class for the common property which would be ordin-
arily assumed. This is the reason for the definitions
we have adopted, and this is the source of the apparent
paradoxes. No harm is done if there are such common
properties as language assumes, since we do not deny
them, but merely abstain from asserting them. But
if there are not such common properties in any given
case, then our method has secured us against error.
In the absence of special knowledge, therefore, the
method we have adopted is the only one which is
safe, and which avoids the risk of introducing fictitious
metaphysical entities.

LECTURE V
THE THEORY OF CONTINUITY

THE theory of continuity, with which we shall be
occupied in the present lecture, is, in most of its
refinements and developments, a purely mathematical
subject very beautiful, very important, and very
delightful, but not, strictly speaMng, a part of philo-
sophy. The logical basis of the theory alone belongs
to philosophy, and alone will occupy us to-night.
The way the problem of continuity enters into philo-
sophy is, broadly speaking, the following : Space and
time are treated by Mhgpiatiffo-Ti s as consisting of
points and instants, but they also have a property,
easier to feel than to define, which is called continuity,
and is thought by many philosophers to be destroyed
when they are resolved into points and instants.
Zeno, as we shall see, proved that analysis into points
and instants was impossible if we adhered to the
view that the number of points or instants in a finite
space or time must be finite. Later philosophers,
believing infinite number to be self-contradictory, have
found here an antinomy : Spaces and times could not
consist of a finite number of points and instants, for
such reasons as Zeno's ; they could not consist of
an infinite number of points and instants, because
infinite numbers were supposed to be self-contradictory.
Therefore spaces and times, if real at all, must not be
regarded as composed of points anci instants,

136 SCIENTIFIC METHOD IN PHILOSOPHY

But even when points and instants, as independent
entities, are discarded, as they were by the theory
advocated in our last lecture, the problems of con-
tinuity, as I shall try to show presently, remain, in a
practically unchanged form. Let us therefore, to
begin with, admit points and instants, and consider
the problems in connection with this simpler or at
least more familiar hypothesis.

The argument against continuity, in so far as it
rests upon the supposed difficulties of infinite numbers,
has been disposed of by the positive theory of the
infinite, which will be considered in Lecture VII.
But there remains a feeling of the kind that led
Zeno to the contention that the arrow in its flight is
at rest which suggests that points and instants, even
if they are infinitely numerous, can only give a jerky
motion, a succession of different immobilities, not
the smooth transitions with which the senses have
made us familiar. This feeling is due, I believe, to
a failure to realize imaginatively, as well as abstractly,
the nature of continuous series as they appear in
mathematics. When a theory has been apprehended
logically, there is often a long and serious labour still
required in order to feel it : it is necessary to dwell
upon it, to thrust out from the mind, one by one, the
misleading suggestions of false but more familiar
theories, to acquire the kind of intimacy which, in
the case of a foreign language, would enable us to
think and dream in it, not merely to construct laborious
sentences by the help of grammar and dictionary.
It is, I believe, the absence of this kind of intimacy
which makes many philosophers regard the mathe-
matical doctrine of continuity as an inadequate
explanation of the continuity which we experience
in the world of sense.

THE THEORY OF CONTINUITY 137

In the present lecture, I shall first try to explain
in outline what the mathematical theory of continuity
is in its philosophically important essentials. The
application to actual space and time will not be in
question to begin with. I do not see any reason to
suppose that the points and instants which mathe-
maticians introduce in dealing with space and time are
actual physically existing entities, but I do see reason
to suppose that the continuity of actual space and
time may be more or less analogous to mathematical
continuity. The theory of mathematical continuity
is an abstract logical theory, not dependent for its
validity upon any properties of actual space and time.
What is claimed for it is that, when it is understood,
certain characteristics of space and time, previously
very hard to analyse, are found not to present any
logical difficulty. What we know empirically about
space and time is insufficient to enable us to decide
between various mathematically possible alternatives,
but these alternatives are all fully intelligible and
fully adequate to the observed facts. For the present,
however, it will be well to forget space and time and
the continuity of sensible change, in order to return
to these topics equipped with the weapons provided
by the abstract theory of continuity.

Continuity, in mathematics, is a property only
possible to a series of terms, i.e. to terms arranged in
an order, so that we can say of any two that one comes
before the other. Numbers in order of magnitude, the
points on a line from left to right, the moments of
time from earlier to later, are instances of series. The
notion of order, which is here introduced, is one which
is not required in the theory of cardinal number.
It is possible to know that two classes have the same
number of terms without knowing any order in which

138 SCIENTIFIC METHOD IN PHILOSOPHY

they are to be taken. We have an instance of this in
such a case as English husbands and English wives :
we can see that there must be the same number of
husbands as of wives, without having to arrange them
in a series. But continuity, which we are now to
consider, is essentially a property of an order: it
does not belong to a set of terms in themselves, but
only to a set in a certain order. A set of terms which
can be arranged in one order can always also be arranged
in other orders, and a set of terms which can be arranged
in a continuous order can always be arranged in orders
which are not continuous. Thus the essence of con-
tinuity must not be sought in the nature of the set
of terms, but in the nature of their arrangement in a
series.

Mathematicians have distinguished different degrees
of continuity, and have confined the word " con-
tinuous," for technical purposes, to series having a
certain high degree of continuity. But for philoso-
phical purposes, all that is important in continuity is
introduced by the lowest degree of continuity, which
is called "compactness." A series is called
"compact" when no two terms are consecutive,
but between any two there are others. One of the
simplest examples of a compact series is the series
of fractions in order of magnitude. Given any two
fractions, however near together, there are other
fractions greater than the one and smaller than the
other, and therefore no two fractions are consecutive.
There is no fraction, for example, which is next after J :
if we choose some fraction which is very little greater
Ibs* i sa Y iVb* TO can find others, such as J-, which
are nearer to . Thus between any two fractions,
however little they differ, there are an infinite "number
of other fractions. Mathematical space and tjm$

THE THEORY OF CONTINUITY 139

also have this property of compactness, though whether
actual space and tune have it is a further question,
dependent upon empirical evidence, and probably
incapable of being answered with certainty.

In the case of abstract objects such as fractions, it
is perhaps not very difficult to realize the logical
possibility of their forming a compact series. The
difficulties that might be felt are those of infinity, for
in a compact series the number of terms between
any two given terms must be infinite. But when these
difficulties have been solved, the mere compactness in .
itself offers no great obstacle to the imagination. In
more concrete cases, however, such as motion, com-
pactness becomes much more repugnant to our habits
of thought. It will therefore be desirable to consider
explicitly the mathematical account of motion, with
a view to making its logical possibility felt. The
mathematical account of motion is perhaps artificially
simplified when regarded as describing what actually
occurs in the physical world; but what actually
occurs must be capable, by a certain amount of logical
manipulation, of being brought within the scope of
the mathematical account, and must, in its analysis,
raise just such problems as are raised in their simplest
form by this account. Neglecting, therefore, for the
present, the question of its physical adequacy, let
us devote ourselves merely to considering its possibility
as a formal statement of the nature of motion.

In order to simplify our problem as much as possible,
let us imagine a tiny speck of light moving along a
scale. What do we mean by saying that the motion
is continuous ? It is not necessary for our purposes
to consider the whole of what the mathematician
means by this statement : only part of what he means
is philosophically important. One part of what he

140 SCIENTIFIC METHOD IN PHILOSOPHY

means is that, if we consider any two positions of
the speck occupied at any two instants, there will be
other intermediate positions occupied at intermediate
instants. However near together we take the two
positions, the speck will not jump suddenly from
the one to the other, but will pass through an infinite
number of other positions on the way. Every dis-
tance, however small, is traversed by passing through
all the infinite series of positions between the two ends
of the distance.

But at this point imagination suggests that we may
describe the continuity of motion by saying that the
speck always passes from one position at one instant
to the next position at the next instant. As soon as
we say this or imagine it, we faJl into error, because
there is no next point or next instant. If there were,
we should find Zeno's paradoxes, in some form, un?
avoidable, as will appear in our next lecture. One
simple, paradox may serve as an illustration. If our
speck is in motion along the scale throughout the
whole of a certain time, it cannot be at the same point
at two consecutive instants. But it cannot, from one
instant to the next, travel further than from one point
to the next, for if it did, there would be no instant at
which it was in the positions intermediate between
that at the first instant and that at the next, and
we agreed that the continuity of motion excludes the
possibility of such sudden jumps. It follows that our
speck must, so long as it moves, pass from one point
at one instant to the next point at the next instant.
Thus there will be just one perfectly definite velocity
with which all motions must take place : no motion
can be faster than this, and no motion can be slower.
Since this conclusion is false, we must reject the hypo-
thesis upon which it is based, namely that there are

THE THEORY OF CONTINUITY 141

consecutive points and instants. 1 Hence the con-
tinuity of motion must not be supposed to consist
in a body's occupying consecutive positions at con-
secutive times.

The difficulty to imagination lies chiefly, I think, in
keeping out the suggestion of infinitesimal distances
and times. Suppose we halve a given distance, and
then halve the half, and so on, we can continue the
process as long as we please, and the longer we con-
tinue it, the smaller the resulting distance becomes.
This infinite divisibility seems, at first sight, to imply
that there are infinitesimal distances, i.e. distances
so small that any finite fraction of an inch would be
greater. This, however, is an error. The continued
bisection of our distance, though it gives us continually
smaller distances, gives us always finite distances. If
pur original distance was an inch, we reach successively
half an inch, a quarter of an inch, an eighth, a six-
teenth, and so on; but every one of this infinite
series of diminishing distances is finite, " But/ 1 it
may be said, " in the end the distance will grow infini-
tesimal." No, because there is no end. The process
of bisection is one which can, theoretically, be carried
on for ever, without any last term being attained.
Thus infinite divisibility of distances, which must be
admitted, does not imply that there are distances so
small that any finite distance would be larger.

It is easy, in this kind of question, to fall into an
elementary logical blunder. Given any finite dis-
tance, we can find a smaller distance ; this may be
expressed in the ambiguous fonn " there is a distance
smaller than any finite distance." But if this is then

i The above paradox is essentially the same as Zeno's
argument of the stadium which will be considered in our next
lecture.

142 SCIENTIFIC METHOD IN PHILOSOPHY

interpreted as meaning " there is a distance such that,
whatever finite distance may be chosen, the distance
in question is smaller," then the statement is false.
Common language is ill adapted to expressing matters
of this kind, and philosophers who have been dependent
on it have frequently been misled by it.

In a continuous motion, then, we shall say that at any
given instant the moving body occupies a certain posi-
tion, and at other instants it occupies other positions ;
the interval between any two instants and between
any two positions is always finite, but the continuity
of the motion is shown in the fact that, however 'near
togetherwe take the two positions and the two instants,
there are an infinite number of positions still nearer
together, which are occupied at instants that are also
still nearer together. The moving body never jumps
from one position to another, but always passes by
a gradual transition through an infinite number of
intermediaries. At a given instant, it is where it is,
like Zeno's arrow ; * but we cannot say that it is at
rest at the instant, since the instant does not last for
a finite time, and there is not a beginning and end
of the instant with an interval between them. Rest
consists in being in the same position at all the instants
throughout a certain finite period, however short ; it
does not consist simply in a body's being where it is
at a given instant. This whole theory, as is obvious,
depends upon the nature of compact series, and
demands, for its full comprehension, that compact
series should have become familiar and easy to the
imagination as well as to deliberate thought.

What is required may be expressed in mathematical
language by saying that the position of a moving body
must be a continuous function of the time. To define
See next lecture.

THE THEORY OF CONTINUITY 143

accurately what this means, we proceed as follows.
Consider a particle which, at the moment t, is at the

- '

point P. Choose now any small portion PiP 2
of the path of the particle, this portion being
one which contains P. We say then that, if the
motion of the particle is continuous at the time *,
it must be possible to find two instants ^, t^, one
earlier than t and one later, such that throughout
the whole time from ^ to 2 (both included), the
particle lies between P! and P 2 . And we say that
this must still hold however small we make the portion
P! P a . When this is the case, we say that the motion
is continuous at the time t ; and when the motion is.
continuous at all times, we say that the motion as a
whole is continuous. It is obvious that if the particle
were to jump suddenly from P to some other point
Q, our definition would fail for all intervals PI PS
which were too small to include Q. Thus our definition
affords an analysis of the continuity of motion, while
admitting points and instants and denying infinitesimal
distances in space or periods in time.

Philosophers, mostly in ignorance of the mathe-
matician's analysis, have adopted other and more
heroic methods of dealing with the prima fade diffi-
culties of continuous motion. A typical and recent
example of philosophic theories of motion is afforded
by Bergson, whose views on this subject I have
examined elsewhere. 1

Apart from definite arguments, there are certain
feelings, rather than reasons, which stand in the way
of an acceptance of the mathematical account of
* Monist, July 1912, pp. 337-341.

144 SCIENTIFIC METHOD IN PHILOSOPHY

motion. To begin with, if a body is moving at all
fast, we see its motion just as we see its colour. A
slow motion, like that of the hour-hand of a watch,
is only known in the way which mathematics would
lead us to expect, namely by observing a change of
position after a lapse of time ; but, when we observe
the motion of the second-hand, we do not merely
see first one position and then another we see some-
thing as directly sensible as colour. What is this
something that we see, and that we call visible motion ?
Whatever it is, it is not the successive occupation of
successive positions : something beyond the mathe-
matical theory of motion is required to account for it.
Opponents of the mathematical theory emphasize this
fact. "Your theory," they say, "may be very
logical, and might apply admirably to some other
world ; but in this actual world, actual motions are
quite different from what your theory would declare
them to be, and require, therefore, some different
philosophy from yours for their adequate explanation."

The objection thus raised is one which I have no
wish to underrate, but I believe it can be fully answered
without departing from the methods and the outlook
which have led to the mathematical theory of motion.
Let us, however, first try to state the objection more
fully.

If the mathematical theory is adequate, nothing
happens when a body moves except that it is in different
places at different times. But in this sense the hour-
hand and the second-hand are equally in motion, yet
in the second-hand there is something perceptible to
our senses which is absent in the hour-hand. We can
see, at each moment, that the second-hand is moving,
which is different from seeing it first in one place and
then in another. This seems to involve our seeing

THE THEORY OF CONTINUITY 145

it simultaneously in a number of places, although it
must also involve our seeing that it is in some of these
places earlier than in others. If, for example, I
move my hand quickly from left to right, you seem
to see the whole movement at once, in spite of the
fact that you know it begins at the left and ends at
the right. It is this kind of consideration, I think,
which leads Bergson and many others to regard a
movement as really one indivisible whole, not the
series of separate states imagined by the mathe-
matician.

To this objection there are three supplementary
answers, physiological, psychological, and logical. We
will consider them successively.

(i) The physiological answer merely shows that, if
the physical world is what the mathmatician supposes,
its sensible appearance may nevertheless be expected
to be what it is. The aim of this answer is thus the
modest one of showing that the mathematical account
is not impossible as applied to the physical world ; it
does not even attempt to show that this account is
necessary, or that an analogous account applies in
psychology.

When any nerve is stimulated, so as to cause a
sensation, the sensation does not cease instantaneously
with the cessation of the stimulus, but dies away in a
short finite time. A flash of lightning, brief as it is
to our sight, is briefer still as a physical phenomenon :
we continue to see it for a few moments after the light-
waves have ceased to strike the eye. Thus in the
case of a physical motion, if it is sufficiently swift, we
shall actually at one instant see the moving body
throughout a finite portion of its course, and not
only at the exact spot where it is at that instant.
Sensations, however, as they die away, grow gradually

10

I 4 6 SCIENTIFIC METHOD IN PHILOSOPHY

fainter ; tnus the sensation due to a stimulus which
is recently past is not exactly like the sensation due
to a present stimulus. It follows from this that,
when we see a rapid motion, we shall not only see a
number of positions of the moving body simultaneously,
but we shall see them with different degrees of intensity
the present position most vividly, and the others
with diminishing vividness, mvril sensation fades
away into immediate memory. This state of things
accounts fully for the perception of motion. A motion
is perceived, not merely inferred, when it is sufficiently
swift for many positions to be sensible at one time ;
and the earlier and later parts of one perceived motion
are distinguished by the less and greater vividness of
the sensations.

This answer shows that physiology can account for
our perception of motion. But physiology, in speaking
of stimulus and sense-organs and a physical motion
distinct from the immediate object of sense, is assuming
the truth of physics, and is thus only capable of show-
ing the physical account to be possible, not of showing
it to be necessary. This consideration brings us to
the psychological answer.

(2) The psychological answer to our difficulty about
motion is part of a vast theory, not yet worked out,
and only capable, at present, of being vaguely outlined.
We considered this theory in the third and fourth
lectures ; for the present, a mere sketch of its applica-
tion to our present problem must suffice. The world of
physics, which was assumed in the physiological
answer, is obviously inferred from what is given in
sensation; yet as soon as we seriously consider what is
actually given in sensation, we find it apparently very
different from the world of physics. The question is
thus forced upon us : Is the inference from sense to

THE THEORY OF CONTINUITY 147

physics a valid one ? I believe the answer to be
affirmative, for reasons which I suggested in the third
and fourth lectures ; but the answer cannot be either
short or easy. It consists, broadly speaking, in show-
ing that, although the particles, points, and instants
with which physics operates are not themselves
given in experience, and are very likely not actually
existing things, yet, out of the materials provided in
sensation, together with other particulars structurally
similar to these materials, it is possible to make logical
constructions having the mathematical properties
which physics assigns to particles, points, and instants.
If this can be done, then all the propositions of physics
can be translated, by a sort of dictionary, into proposi-
tions about the kinds of objects which are given in
sensation.

Applying these general considerations to the case
of motion, we find that, even within the sphere of
immediate sense-data, it is necessary, or at any rate
more consonant with the facts than any other equally
simple view, to distinguish instantaneous states of
objects, and to regard such states as forming a compact
series. Let us consider a body which is moving swiftly
enough for its motion to be perceptible, and long enough
for its motion to be not wholly comprised in one
sensation. Then, in spite of the fact that we see
a finite extent of the motion at one instant, the extent
which we see at one instant is different from that
which we see at another. Thus we are brought
back, after all, to a series of momentary views of the
moving body, and this series will be compact, like
the former physical series of points. In fact, though
the terms of the series seem different, the mathematical
character of the series is unchanged, and the whole
mathematical theory of motion will apply to it verbatim.

148 SCIENTIFIC METHOD IN PHILOSOPHY

When we are considering the actual data of sensa-
tion in this connection, it is important to realize that
two sense-data may be, and must sometimes be, really
different when we cannot perceive any difference
between them. An old but conclusive reason for
believing this was emphasized by PoincarS. 1 In all
cases of sense-data capable of gradual change, we
may find one sense-datum indistinguishable from
another, and that other indistinguishable from a
third, while yet the first and third are quite easily
distinguishable. Suppose, for example, a person with
his eyes shut is holding a weight in his hand, and
someone noiselessly adds a small extra weight. If
the extra weight is small enough, no difference will be
perceived in the sensation. After a time, another
small extra weight may be added, and still no change
will be perceived; but if both extra weights had
been added at once, it may be that the change would
be quite easily perceptible. Or, again, take shades
of colour. It would be easy to find three stuffs of
such closely similar shades that no difference could be
perceived between the first and second, nor yet between
the second and third, while yet the first and third
would be distinguishable. In such a case, the second
shade cannot be the same as the first, or it would be
distinguishable from the third ; nor the same as the
third, or it would be distinguishable from the first.
It must, therefore, though indistinguishable from
both, be really intermediate between them.

Such considerations as the above show that, although
we cannot distinguish sense-data unless they differ
by more than a certain amount, it is perfectly reason-
able to suppose that sense-data of a given kind, such

1 " Le continu math&natique/' Revue de Mttaphysique et fo
Morale, vol. i. p. 29.

THE THEORY OF CONTINUITY 149

as weights or colours, really form a compact series.
The objections which may be brought from a psycho-
logical point of view against the mathematical theory
of motion are not, therefore, objections to this theory
properly understood, but only to a quite unnecessary
assumption of simplicity in the momentary object
of sense. Of the immediate object of sense, in the case
of a visible motion, we may say that at each instant
it is in all the positions which remain sensible at that
instant ; but this set of positions changes continuously
from moment to moment, and is amenable to exactly
the same mathematical treatment as if it were a mere
point. When we assert that some mathematical
account of phenomena is correct, all that we primarily
assert is that something definable in terms of the crude
phenomena satisfies our formulae ; and in this sense
the mathematical theory of motion is applicable to
the data of sensation as well as to the supposed par-
ticles of abstract physics.

There are a number of distinct questions which are
apt to be confused when the mathematical con-
tinuum is said to be inadequate to the facts of sense.
We may state these, in order of diminishing generality,
as follows :

(a) Are series possessing mathematical con-
tinuity logically possible ?

(6) Assuming that they are possible logically, are
they not impossible as applied to actual sense-
data, because, among actual sense-data, there are
no such fixed mutually external terms as are to
be found, e.g. in the series of fractions ?

(c) Does not the assumption of points and
instants make the whole mathematical account
fictitious ?

ISO SCIENTIFIC METHOD IN PHILOSOPHY

() Finally, assuming that all these objections
have been answered, is there, in actual empirical
fact, any sufficient reason to believe the world of
sense continuous ?

Let us consider these questions in succession.

(a) The question of the logical possibility of the
mathematical continuum turns partly on the ele-
mentary misunderstandings we considered at the
beginning of the present lecture, partly on the possi-
bility of the mathematical infinite, which will occupy
our next two lectures, and partly on the logical form
of the answer to the Bergsonian objection which we
stated a few minutes ago. I shall say no more on
this topic at present, since it is desirable first to com-
plete the psychological answer.

(6) The question whether sense data are composed of
mutually external units is not one which can be decided
by empirical evidence. It is often urged that, as a
matter of immediate experience, the sensible flux is
devoid of divisions, and is falsified by the dissections
of the intellect. Now I have no wish to argue that
this view is contrary to immediate experience : I wish
only to maintain that it is essentially incapable of
being proved by immediate experience. As we saw,
there must be among sense-data differences so slight
as to be imperceptible : the fact that sense-data are
immediately given does not mean that their differences
also must be immediately given (though they may
be). Suppose, for example, a coloured surface on
which the colour changes gradually so gradually
that the difference of colour in two very neighbouring
portions is imperceptible, while the difference between
more widely separated portions is quite noticeable.
The effect produced, in such a case, will be precisely

THE THEORY OF CONTINUITY 151

that of " interpenetration," of transition which is
not a matter of discrete units. And since it tends to
be supposed that the colours, being immediate data,
must appear different if they are different, it seems
easily to follow that "interpenetration" must be
the ultimately right account. But this does not follow.
It is unconsciously assumed, as a premiss for a reductio
ai dbsurdum of the analytic view, that, if A and B
are immediate data, and A differs from B, then the
fact that they differ must also be an immediate datum.
It is difficult to say how this assumption arose, but I
think it is to be connected with the confusion between
" acquaintance " and " knowledge about." Acquaint-
ance, which is what we derive from sense, does not,
theoretically at least, imply even the smallest " know-
ledge about," i.e. it does not imply knowledge of any
proposition concerning the object with which we are
acquainted. It is a mistake to speak as if acquaint-
ance had degrees : there is merely acquaintance and
non-acquaintance. When we speak of becoming
" better acquainted," as for instance with a person,
what we must mean is, becoming acquainted with
more parts of a certain whole ; but the acquaintance
with each part is either complete or non-existent.
Thus it is a mistake to say that if we were perfectly
acquainted with an object we should know all about
it. " Knowledge about " is knowledge of proposi-
tions, which is not involved necessarily in acquaint-
ance with the constituents of the propositions. To
know that two shades of colour are different is know-
ledge about them ; hence acquaintance with the two
shades does not in any way necessitate the knowledge
that they are different.

From what has just been said it follows that the
nature of sense-data cannot be validly used to prove

152 SCIENTIFIC METHOD IN PHILOSOPHY

that they are not composed of mutually external
units. It may be admitted, on the other hand, that
nothing in their empirical character specially necessi-
tates the view that they are composed of mutually
external units. This view, if it is held, must be held
on logical, not on empirical grounds. I believe that
the logical grounds axe adequate to the conclusion.
They rest, at bottom, upon the impossibility of ex-
plaining complexity without assuming constituents.
It is undeniable that the visual field, for example,
is complex ; and so far as I can see, there is always
self-contradiction in the theories which, while admitting
this complexity, attempt to deny that it results from
a combination of mutually external units. But to
pursue this topic would lead us too far from our theme,
and I shall therefore say no more about it at present.

(c) It is sometimes urged that the mathematical
account of motion is rendered fictitious by its assump^
tion of points and instants. Now there are here two
different questions to be distinguished. There is the
question of absolute or relative space and time, and
there is the question whether what occupies space
and time must be composed of elements which have
no extension or duration. And each of these ques-
tions in turn may take two forms, namely : (a) is
the hypothesis consistent with the facts and with
logic ? 08) is it necessitated by the facts or by logic ?
I wish to answer, in each case, yes to the first form
of the question, and no to the second. But in any
case the mathematical account of motion will not be
fictitious, provided a right interpretation is given
to the words " point " and " instant." A few words
on each alternative will serve to make this clear.

Formally, mathematics adopts an absolute theory
of space and time, ie. it assumes that, besides the

THE THEORY OF CONTINUITY 153

things which are in space and time, there are also
entities, called "points" and "instants/' which
axe occupied by things. This view, however, though
advocated by Newton, has long been regarded by
mathematicians as merely a convenient fiction. There
is, so far as I can see, no conceivable evidence either
for or against it. It is logically possible, and it is
consistent with the facts. But the facts are also
consistent with the denial of spatial and temporal
entities over and above things with spatial and tem-
poral relations. Hence, in accordance with Occam's
razor, we shall do well to abstain from either assuming
or denying points and instants. This means, so far
as practical working out is concerned, that we adopt the
relational theory ; for in practice the refusal to assume
points and instants has the same effect as the denial
of them. But in strict theory the two are quite
different, since the denial introduces an element of
unverifiable dogma which is wholly absent when we
merely refrain from the assertion. Thus, although
we shall derive points and instants from things, we
shall leave the bare possibility open that they may
also have an independent existence as simple entities.

We come now to the question whether the things
in space and time are to be conceived as composed of
elements without extension or duration, i.e. of elements
which only occupy a point and an instant. Physics,
formally, assumes in its differential equations that
things consist of elements which occupy only a point
at each instant, but persist throughout time. For
reasons explained in Lecture IV, the persistence of
things through time is to be regarded as the
formal result of a logical construction, not as necessarily
implying any actual persistence. The same motives,
in fact, which lead to the division of things into point-

154 SCIENTIFIC METHOD IN PHILOSOPHY

particles, ought presumably to lead to their division
into instant-particles, so that the ultimate formal
constituent of the matter in physics will be a point-
instant-particle. But such objects, as well as the
particles of physics, are not data. The same economy
of hypothesis, which dictates the practical adoption
of a relative rather than an absolute space and time,
also dictates the practical adoption of material elements
which have a finite extension and duration. Since,
as we saw in Lecture IV, points and instants can be
constructed as logical functions of such elements, the
mathematical account of motion, in which a particle
passes continuously through a continuous series of
points, can be interpreted in a form which assumes
only elements which agree with our actual data in
having a finite extension and duration. Thus, so
far as the use of points and instants is concerned, the
mathematical account of motion can be freed from
the charge of employing fictions.

(f} But we must now face the question : Is there,
in actual empirical fact, any sufficient reason to
believe the world of sense continuous ? The answer
here must, I think, be in the negative. We may
say that the hypothesis of continuity is perfectly
consistent with the facts and with logic, and that it
is technically simpler than any other tenable hypo-
thesis. But since our powers of discrimination among
very similar sensible objects are not infinitely precise,
it is quite impossible to decide between different
theories which only differ in regard to what is below
the margin of dfecrimination. If, for example, a
coloured surface which we see consists of a finite
number of very small surfaces, and if a motion which
we see consists, like a cinematograph, of a large finite
number of successive positions, there will be nothing

THE THEORY OF CONTINUITY 155

empirically discoverable to show that objects of sense
are not continuous. In what is called experienced con-
tinuity, such as is said to be given in sense, there is a
large negative element: absence of perception of
difference occurs in cases which are thought to give
perception of absence of difference. When, for
example, we cannot distinguish a colour A from a
colour B, nor a colour B from a colour C, but can
distinguish A from C, the indistinguishabiHty is a
purely negative fact, namely, that we do not perceive
a difference. Even in regard to immediate data,
this is no reason for denying that there is a difference.
Thus, if we see a coloured surface whose colour changes
gradually, its sensible appearance if the change is
continuous will be indistinguishable from what it
would be if the change were by small finite jumps. If
this is true, as it seems to be, it follows that there can
never be any empirical evidence to demonstrate that
the sensible world is continuous, and not a collection
of a very large finite number of dements of which each
differs from its neighbour in a finite though very small
degree. The continuity of space and time, the infinite
number of different shades in the spectrum, and so
on, are all in the nature of unverifiable hypotheses
perfectly possible logically, perfectly consistent
with the known facts, and simpler technically than
any other tenable hypotheses, but not the sole hypo-
theses which are logically and empirically adequate.

If a relational theory of instants is constructed, in
which an " instant " is defined as a group of events
simultaneous with each other and not all simultaneous
with any event outside the group, then if our resulting
series of instants is to be compact, it must be possible,
if x wholly precedes y , to find an event z, simultaneous
with part of x, which wholly precedes some event

156 SCIENTIFIC METHOD IN PHILOSOPHY

which wholly precedes y. Now this requires that the
number of events concerned should be infinite in any
finite period of time. If this is to be the case in the
world of one man's sense-data, and if each sense-
datum is to have not less than a certain finite temporal
extension, it will be necessary to assume that we always
have an infinite number of sense-data simultaneous
with any given sense-datum. Applying similar con-
siderations to space, and assuming that sense-data
are to have not less than a certain spatial extension,
it will be necessary to suppose that an infinite number
of sense-data overlap spatially with any given sense-
datum. This hypothesis is possible, if we suppose a
single sense-datum, e.g. in sight, to be a finite surface,
enclosing other surfaces which are also single sense-
data. But there are difficulties in such a hypothesis,
and I do not think that these difficulties could be
successfully met. If they cannot, we must do one
of two things : either declare that the world of one
man's sense-data is not continuous, or else refuse to
admit that there is any lower limit to the duration
and extension of a single sense-datum. The latter
hypothesis seems untenable, so that we are apparently
forced to conclude that the space of sense-data is
not continuous ; but that does not prevent us from
admitting that sense-data have parts which are not
sense-data, and that the space of these parts may be
continuous. The logical analysis we have been con-
sidering provides the apparatus for dealing with the
various hypotheses, and the empirical decision between
them is a problem for the psychologist.

(3) We have now to consider the logical answer to the
alleged difficulties of the mathematical theory of
motion, or rather to the positive theory which is
urged on the other side. The view urged explicitly

THE THEORY OF CONTINUITY 157

by Bergson, and implied in the doctrines of many
philosophers, is, that a motion is something indivisible,
not validly analysable into a series of states. This is
part of a much more general doctrine, which holds
that analysis always falsifies, because the parts of a
complex whole are different, as combined in that whole,
from what they would otherwise be. It is very difficult
to state this doctrine in any form which has a precise
meaning. Often arguments are used which have no
bearing whatever upon the question. It is urged,
for example, that when a man becomes a father,
his nature is altered by the new relation in which he
finds himself, so that he is not strictly identical with
the man who was previously not a father. This may
be true, but it is a causal psychological fact, not a
logical fact. The doctrine would require that a man
who is a father cannot be strictly identical with a
man who is a son, because he is modified in one way
by the relation of fatherhood and in another by that
of sonship. In fact, we may give a precise statement
of the doctrine we are combating in the form : There
can never be two facts concerning the same thing. A
fact concerning a thing always is or involves a relation
to one or more entities ; thus two facts concerning the
same thing would involve two relations of the same
thing. But the doctrine in question holds that a thing
is so modified by its relations that it cannot be the same
in one relation as in another. Hence, if this doctrine
is true, there can never be more than one fact con-
cerning any one thing. I do not think the philosophers
in question have realized that this is the precise state-
ment of the view they advocate, because in this form
the view is so contrary to plain truth that its falsehood
is evident as soon as it is stated. The discussion of
this question, however, involves so many logical

158 SCIENTIFIC METHOD IN PHILOSOPHY

subtleties, and is so beset with difficulties, that I shall
not pursue it further at present.

When once the above general doctrine is rejected,
it is obvious that, where there is change, there must
be a succession of states. There cannot be change
and motion is only a particular case of change unless
there is something different at one time from what
there is at some other time. Change, therefore, must
involve relations and complexity, and must demand
analysis. So long as our analysis has only gone as
far as other smaller changes, it is not complete ; if
it is to be complete, it must end with terms that are
not changes, but are related by a relation of earlier
and later. In the case of changes which appear
continuous, 'such as motions, it seems to be impos-
sible to find anything other than change so long as
we deal with finite periods of time, however short.
We are thus driven back, by the logical necessities
of the case, to the conception of instants without
duration, or at any rate without any duration which
even the most delicate instruments can reveal. This
conception, though it can be made to seem difficult, is
really easier than any other that the facts allow. It
is a kind of logical framework into which any tenable
theory must fit not necessarily itself the statement
of the crude facts, but a form in which statements
which are true of the crude facts can be i&ade by a
suitable interpretation. The direct consideration of
the crude facts of the physical world has been under-
taken in earlier lectures ; in the present lecture, we
have only been concerned to show that nothing in
the crude facts is inconsistent with the mathematical
doctrine of continuity, or demands a continuity of
a radically different kind from that of mathematical
motion.

LECTURE VI

THE PROBLEM OF INFINITY
CONSIDERED HISTORICALLY

IT will be remembered that, when we enumerated the
grounds upon which the reality of the sensible world
has been questioned, one of those mentioned was the
supposed impossibility of infinity and continuity. In
view of our earlier discussion of physics, it would seem
that no conclusive empirical evidence exists in favour
of infinity or continuity in objects of sense or in matter.
Nevertheless, the explanation which assumes infinity
and continuity remains incomparably easier and more
natural, from a scientific point of view, than any other,
and since Georg Cantor has shown that the supposed
contraditions are illusory, there is no longer any reason
to struggle after a finitist explanation of the world.

The supposed difficulties of continuity all have their
source in the fact that a continuous series must have
an infinite number of terms, and are in fact difficulties
concerning infinity. Hence, in freeing the infinite
from contradiction, we are at the same time showing
the logical possibility of continuity as assumed in
science.

The kind of way in which infinity has been used to
discredit the world of sense may be illustrated by
Kant's first two antinomies. In the first, the thesis
states : " The world has a beginning in time, and as

160 SCIENTIFIC METHOD IN PHILOSOPHY

regards space is enclosed within limits " ; the anti-
thesis states : " The world has no beginning and no
limits in space, but is infinite in respect of both time
and space." Kant professes to prove both these
propositions, whereas, if what we have said on modern
logic has any truth, it must be impossible to prove
either. In order, however, to rescue the world of
sense, it is enough to destroy the proof of one of
the two. For our present purpose, it is the proof that
the world is finite that interests us. Kant's argument
as regards space here rests upon his argument as
regards time. We need therefore only examine the
argument as regards time. What he says is as follows :

" For let us assume that the world has no beginning
as regards time, so that up to every given instant an
eternity has elapsed, and therefore an infinite series of
successive states of the things in the world has passed
by. But the infinity of a series consists just in this,
that it can never be completed by successive syn-
thesis. Therefore an infinite past world-series is
impossible, and accordingly a .beginning of the world
is a necessary condition of its existence ; which was
the first thing to be proved."

Many different criticisms might be passed on this
argument, but we will content ourselves with a bare
TninjTnnTn , To begin with, it is a mistake to define
the infinity of a series as " impossibility of completion
by successive synthesis." The notion of infinity, as
we shall see in the next lecture, is primarily a property
of dosses, and only derivatively applicable to series ;
classes which are infinite are given all at once by the
defining properly of their members, so that there is
no question of " completion " or of " successive syn-
thesis." And the word "synthesis," by suggesting
the mental activity of synthesizing, introduces, more

THE PROBLEM OF INFINITY 161

or less surreptitiously, that reference to mind by which
all Kant's philosophy was infected. In the second
place, when Kant says that an infinite series can
" never " be completed by successive synthesis, all
that he has even conceivably a right to say is that it
cannot be completed in a finite time. Thus what he
really proves is, at most, that if the world had no
beginning, it must have already existed for an infinite
time. This, however, is a very poor conclusion, by
no means suitable for his purposes. And with this
result we might, if we chose, take leave of the first
antinomy.

It is worth while, however, to consider how Kant
came to make such an elementary blunder. What
happened in his imagination was obviously something
like this : Starting from the present and going back-
wards in time, we have, if the world had no beginning,
an infinite series of events. As we see from the word
"synthesis," he imagined a mind trying to grasp
these successively, in the reverse order to that in
which they had occurred, i.e. going from the present
backwards. This series is obviously one which has
no end. But the series of events up to the present
has an end, since it ends with the present. Owing to
the inveterate subjectivism of his mental habits, he
failed to notice that he had reversed the sense of the
series by substituting backward synthesis for forward
happening, and thus he supposed that it was necessary
to identify the mental series, which had no end, with
the physical series, which had an end but no beginning.
It was this mistake, I think, which, operating
unconsciously, led M to attribute validity to a singu-
larly flimsy piece of fallacious reasoning.

The second antimony illustrates the dependence of
the problem of continuity upon that of infinity. The

XI

162 SCIENTIFIC METHOD IN PHILOSOPHY

thesis states : " Every complex substance in the
world consists of simple parts, and -there exists every-
where nothing but the simple or what is composed
of it." The antithesis states : " No complex thing
in the world consists of simple parts, and everywhere in
it there exists nothing simple." Here, as before, the
proofs of both thesis and antithesis are open to criti-
cism, but for the purpose of vindicating physics and
the world of sense it is enough to find a fallacy in
one of the proofs. We will choose for this purpose
the proof of the antithesis, which begins as follows :

" Assume that a complex thing (as substance) con-
sists of simple parts. Since all external relation, and
therefore all composition out of substances, is only
possible in space, the space occupied by a complex
thing must consist of as many parts as the thing con-
sists of. Now space does not consist of simple parts,
but of spaces."

The rest of his argument need not concern us, for
the nerve of the proof lies in the one statement :
" Space does not consist of simple parts, but of spaces."
This is like Bergson's objection to " the absurd pro-
position that motion is made up of immobilities."
Kant does not tell us why he holds that a space must
consist of spaces rather than of simple parts. Geo-
metry regards space as made up of points, which are
simple ; and although, as we have seen, this view is
not scientifically or logically necessary, it remains
prima facie possible, and its mere possibility is enough
to vitiate Kant's argument. For, if his proof of the
thesis of the antinomy were valid, and if the antithesis
could only be avoided by assuming points, then the
antinomy itself would afford a conclusive reason in
favour of points. Why, then, did Kant tTiinlr it im-
possible that space should be composed of points ?

THE PROBLEM OF INFINITY 163

I think two considerations probably influenced him.
In the first place, the essential thing about space is
spatial order, and mere points, by themselves, will
not account for spatial order. It is obvious that his
argument assumes absolute space ; but it is spatial
relations that are alone important, and they cannot
be reduced to points. This ground for his view
depends, therefore, upon his ignorance of the logical
theory of order and his oscillations between absolute
and relative space. But there is also another ground
for his opinion, which is more relevant to our present
topic. This is the ground derived from infinite divisi-
bility. A space may be halved, and then halved again,
and so on ad infinitwn, and at every stage of the pro-
cess the parts are still spaces, not points. In order
to reach points by such a method, it would be necessary
to come to the end of an unending process, which is
impossible. But just as an infinite class can be given
all at once by its defining concept, though it cannot be
reached by successive enumeration, so an infinite
set of points can be given all at once as making up a
line or area or volume, though they can never be
reached by the process of successive division. Thus the
infinite divisibility of space gives no ground for deny-
ing that space is composed of points. Kant does not
give his grounds for this denial, and we can therefore
only conjecture what they were. But the above
two grounds, which we have seen to be fallacious,
seem sufficient to account for his opinion, and we may
therefore conclude that the antithesis of the second
antinomy is unproved.

The above illustration of Kant's antinomies has
only been introduced in order to show the relevance
of the problem of infinity to the problem of the reality
of objects of sense. In the remainder of the present

164 SCIENTIFIC METHOD IN PHILOSOPHY

lecture, I wish to state and explain the problem of
infinity, to show how it arose, and to show the irrele-
vance of all the solutions proposed by philosophers.
In the following lecture, I shall try to explain the
true solution, which has been discovered by the
mathematicians, but nevertheless belongs essentially
to philosophy. The solution is definitive, in the sense
that it entirely satisfies and convinces all who study it
carefully. For over two thousand years the human
intellect was baffled by the problem ; its many failures
and its ultimate success maie this problem peculiarly
apt for the illustration of method.

The problem appears to have first arisen in some such
way as the following. 1 Pythagoras and his followers,
who were interested, like Descartes, in the application
gf number to geometry, adopted in that science more
arithmetical methods than those with which Euclid
has made us familiar. They, or their contemporaries
the atomists, believed, apparently, that space is com-
posed of indivisible points, while time is composed
of indivisible instants.* This belief would not, by
itself, have raised the difficulties which they encoun-
tered, but it was presumably accompanied by another
belief, that the number of points in any finite area
or of instants in any finite period must be finite. I
do not suppose that this latter belief was a conscious
one, because probably no other possibility had occurred
to them. But the belief nevertheless operated, and

* In what concerns the early Greek philosophers, my
knowledge is largely derived from Burnet's valuable work,
Early Greek Philosophy (2nd ed., London, 1908). I have also
been greatly assisted by Mr. D. S. Robertson of Trinity College,
who has supplied the deficiencies of my knowledge of Greek,
and brought important references to my notice.

* Cf . Aristotle, Metaphysics, M. 6, 10806, 18 sqq., and
10836, 8 sqq.

THE PROBLEM OF INFINITY 165

very soon brought them into conflict with facts which
they themselves discovered. Before explaining how
this occurred, however, it is necessary to say one word
in explanation of the phrase " finite number." The
exact explanation is a matter for our next lecture ; for
the present, it must suffice to say that I mean o and
I and 2 and 3 and so one, for ever in other words,
any number that can be obtained by successively
adding ones. This includes all the numbers that
can be expressed by means of our ordinary numerals,
and since such numbers can be made greater and
greater, without ever reaching an unsurpassable
maximum, it is easy to suppose that there are no other
numbers. But this supposition, natural as it is, is
mistaken.

Whether the Pythagoreans themselves believed
space and time to be composed of indivisible points
and instants is a debatable question. 1 It would seem
that the distinction between space -and matter had

z There is some reason to think that the Pythagoreans .
distinguished between discrete and continuous quantity.
G. J. Allman, in his Greek Geometry from Tholes to Euclid.
says (p. 23) : " The Pythagoreans made a fourfold division of
mathematical science, attributing one of its parts to the how
many, ?6 ir6aov, and the other to the how much, r6 tn\\lKw ;
and they assigned to each of these parts a twofold division.
For they said that discrete quantity, or the how many, either
subsists by itself or must be considered with relation to some
other ; but that continued quantity, or the how much, is either
stable or in motion. Hence they affirmed that arithmetic
contemplates that discrete quantity which subsists by itself,
but music that which is related to another ; and that geometry
considers continued quantity so far as it is immovable ; but
astronomy (rip afaupucfa)- contemplates continued quantity
so far as it is of a self-motive nature. (Proclus, ed. Friedlein,
p. 35. As to the distinction between rd mjMxov, continuous,
and TO ir6oov, discrete quantity, see Iambi., in Nicomachi
Geyaseni Arithmeticam introductionem, ed. Tennulius, p. 148.) "

a. P . 4 8-

166 SCIENTIFIC METHOD IN PHILOSOPHY

not yet been clearly made, and that therefore, when
an atomistic view is expressed, it is difficult to decide
whether particles of matter or points of space are
intended. There is an interesting passage z in
Aristotle's Physics* where he says :

" The Pythagoreans all maintained the existence of
the void, and said that it enters into the heaven itself
from the boundless breath, inasmuch as the heaven
breathes in the void also ; and the void differentiates
natures, as if it were a sort of separation of consecu-
tives, and as if it were their differentiation ; and that
this also is what is first in numbers, for it is the void
which differentiates them."

This seems to imply that they regarded matter as
consisting of atoms with empty space in between.
But if so, they must have thought space could be
studied by only paying attention to the atoms, for
otherwise it would be hard to account for their arith-
metical methods in geometry, or for their statement
that " things are numbers."

The difficulty which beset the Pythagoreans in
their attempts to apply numbers arose through their
discovery of incommensurables, and this, in turn, arose
as follows. Pythagoras, as we all learnt in youth,
discovered the proposition that the sum of the squares
on the sides of a right-angled triangle is equal to the
square on the hypotenuse. It is said that he sacrificed
an ox when he discovered this theorem ; if so, the ox
was the first martyr to science. But the theorem,
though it has remained his chief claim to immortality,
was soon found to have a consequence fatal to his

1 Referred to by Burnet, op. dt., p. 120.

iv. f 6. 213$, 22 ; H. Ritter and L. Preller, Historia Philo-
sophies Gr&ca, 8th ed. f Gotha, 1898, p. 75 (this work will be
referred to in future as " R. P.").

THE PROBLEM OF INFINITY 167

whole philosophy. Consider the case of a right-angled
triangle whose two sides are equal, such a triangle
as is formed by two sides of a square and a diagonal.
Here, in virtue of the theorem, the square on the
diagonal is double of the square on either of the sides.
But Pythagoras or his early followers easily proved
that the square of one whole number cannot be double
of the square of another. 1 Thus the length of the
side and the length of the diagonal are incommen-
surable ; that is to say, however small a unit of length
you take, if it is contained an exact number of times
in the side, it is not contained any exact number of
times in the diagonal, and vice versa.

Now this fact might have been assimilated by sortie
philosophies without any great difficulty, but to the
philosophy of Pythagoras it was absolutely fatal.
Pythagoras held that number is the constitutive
essence of all things, yet no two numbers could express
the ratio of the side of a square to the diagonal. It
would seem probable that we may expand his difficulty,
without departing from his thought, by assuming that
he regarded the length of a line as determined by the
number of atoms contained in it a line two inches
long would contain twice as many atoms as a line
one inch long, and so on. But if this were the truth,
then there must be a definite numerical ratio between

1 The Pythagorean proof is roughly as follows. If possible,
let the ratio of the diagonal to the side of a square be m/n,
where m and n are whole numbers having no common factor.
Then we must have w a = 2n*. Now the square of an odd
number is odd, but m a , being equal to 2, is even. Hence
m must be even. But the square of an even number divides
by 4, therefore *, which is half of m*, must be even. There-
fore n must be even. But, since m is even, and m and n have
no common factor, n must be odd. Thus n must be both odd
and even, which is impossible; and therefore the diagonal
and the side cannot have a rational ratio.

168 SCIENTIFIC METHOD IN PHILOSOPHY

any two finite lengths, because it was supposed that
the number of atoms in each, however large, must be
finite. Here there was an insoluble contradiction.
The Pythagoreans, it is said, resolved to keep the
existence of incommensurables a profound secret,
revealed only to a few of the supreme heads of the
sect ; and one of their number, Hippasos of Meta-
pontion, is even said to have been shipwrecked at sea
for impiously disclosing the terrible discovery to
their enemies. It must be remembered that Pytha-
goras was the founder of a new religion as well as the
teacher of a new science : if the science came to be
doubted, the disciples might fall into sin, and perhaps
even eat beans, which according to Pythagoras is as
bad as eating parents' bones.

The problem first raised by the discovery of incom-
mensurables proved, as time went on, to be one of the
most severe and at the same time most far-reaching
problems that have confronted the human intellect
in its endeavour to understand the world. It showed
at once that numerical measurement of lengths, if it
was to be made accurate, must require an arithmetic
more advanced and more difficult than any that the
ancients possessed. They therefore set to work to
reconstruct geometry on a basis which did not assume
the universal possibility of numerical measurement
a reconstruction which, as may be seen in Euclid,
they effected with extraordinary skill and with great
logical acumen. The moderns, under the influence
of Cartesian geometry, have reasserted the universal
possibility of numerical measurement, extending arith-
metic, partly for that purpose, so as to include what
are called " irrational " numbers, which give the ratios
of incommensurable lengths. But although irrational
numbers have long been used without a qualm, it is

THE PROBLEM OF INFINITY 169

only in quite recent years that logically satisfactory
definitions of them have been given. With these
definitions, the first and most obvious form of the
difficulty which confronted the Pythagoreans has
been solved ; but other forms of the difficulty remain
to be considered, and it is these that introduce us
to the problem of infinity in its pure form.

We saw that, accepting the view that a length is
composed of points, the existence of incommensurables
proves that every finite length must contain an infinite
number of points. In other words, if we were to take
away points one by one, we should never have taken
away all the points, however long we continued the
process. The number of points therefore, cannot be
counted, for counting is a process which enumerates
things one by one. The property of being unable to
be counted is characteristic of infinite collections, and
is a source of many of their paradoxical qualities.
So paradoxical are these qualities that until our own
day they were thought to constitute logical contra-
dictions. A long line of philosophers, from Zeno x
to M. Bergson, have based much of their metaphysics
upon the supposed impossibility of infinite collections.
Broadly speaMng, the difficulties were stated by Zeno,
and nothing material was added until we reach Bol-
zano's Paradoxien des Unendlichlen, a little work
written .in 1847-8, and published posthumously in
1851. Intervening attempts to deal with the problem
are futile and negligible. The definitive solution oi
the difficulties is due, not to Bolzano, but to Georg
Cantor, whose work on this subject first appeared in
1882.

1 In regard to Zeno and the Pythagoreans, I have derived
much valuable information and criticism from Mr. P. . B
Jourdain.

170 SCIENTIFIC METHOD IN PHILOSOPHY

In order to understand Zeno, and to realize how
little modern orthodox metaphysics has added to the
achievements of the Greeks, we must consider for a
moment his master Pannenides, in whose interest
the paradoxes were invented. 1 Parmenides expounded
his views in a poem divided into two parts, called
" the way of truth " and " the way of opinion "
like Mr. Bradley's "Appearance" and "Reality,"
except that Paxmenides tells us first about reality
and then about appearance. " The way of opinion,"
in his philosophy, is, broadly speaking, Pytha-
goreanism ; it begins with a warning : " Here I shall
close my trustworthy speech and thought about the
truth. Henceforward learn the opinions of mortals,
giving ear to the deceptive ordering of my words."
What has gone before has been revealed by a goddess,
who tells him what really is. Reality, she says, is
uncreated, indestructible, unchanging, indivisible; it
is " immovable in the bonds of mighty chains, without
beginning and without end ; since coming into being
and passing away have been driven afar, and true
belief has cast them away." The fundamental prin-
ciple of his inquiry is stated in a sentence which would
not be out of place in Hegel : " Thou canst not know
what is not that is impossible nor utter it ; for it
is the same thing that can be thought and that can
be." And again : " It needs must be that what can
be thought and spoken of is ; for it is possible for it
to be, and it is not possible for what is nothing to be."
The impossibility of change follows from this principle ;

So Plato makes Zeno say in the Parmenides, apropos of
his philosophy as a whole; and all internal and external
evidence supports this view.

* " With Parmenides," Hegel says, " philosophizing proper
began." Werke (edition of 1840), vol. xiii. p. 274.

THE PROBLEM OF INFINITY 171

for what is past can be spoken of, and therefore, by
the principle, still is.

The great conception of a reality behind the passing
illusions of sense, a reality one, indivisible, and un-
changing, was thus introduced into Western philosophy
by Parmenides, not, it would seem, for mystical or
religious reasons, but on the basis of a logical argument
as to the impossibility of not-being. All the great
metaphysical systems notably those of Plato, Spinoza,
and Hegel are the outcome of this fundamental
idea. It is difficult to disentangle the truth and the
error in this view. The contention that time is unreal
and that the world of sense is illusory must, I think,
be regarded as based upon fallacious reasoning.
Nevertheless, there is some sense easier to fed than
to state in which time is an unimportant and super-
ficial characteristic of reality. Past and future must
be acknowledged to be as real as the present, and a
certain emancipation from slavery to time is essential
to philosophic thought. The importance of time is
rather practical than theoretical, rather in relation
to our desires than in relation to truth. A truer
image of the world, I think, is obtained by picturing
things as entering into the stream of time from an
eternal world outside, than from a view which regards
time as the devouring tyrant of all that is. Both in
thought and in feeling, to realize the unimportance of
time is the gate of wisdom. But unimportance is
not unreality ; and therefore what we shall have to
say about Zeno's arguments in support of Parmenides
must be mainly critical

The relation of Zeno to Parmenides is explained
by Plato x in the dialogue in which Socrates, as a young
man, learns logical acumen and philosophic dis-
Parmenides, 128 A-D.

SCIENTIFIC METHOD IN PHILOSOPHY

interestedness from their dialectic. I quote from
Jowett's translation :

"I see, Parmenides, said Socrates, that Zeno is
your second self in his writings too; he puts what
you say in another way, and would fain deceive us
into believing that he is telling us what is new. For
you, in your poems, say All is one, and of this you
adduce excellent proofs ; and he on the other hand
says There is no Many ; and on behalf of this he offers
overwhelming evidence. To deceive the world, as
you have done, by saying the same thing in different
ways, one of you affirming the one, and the other
denying the many, is a strain of art beyond the reach
of most of us.

" Yes, Socrates, said Zeno. But although you are
as keen as a Spartan hound in pursuing the track,
you do not quite apprehend the true motive of the
composition, which is not really such an ambitious
work as you imagine ; for what you speak of was an
accident ; I had no serious intention of deceiving the
world. The truth is that these writings of mine were
meant to protect the arguments of Parmenides against
those who scoff at him and show the many ridiculous
and contradictory results which they suppose to
follow from the affirmation of the one. My answer
is an address to the partisans of the many, whose
attack I return with interest by retorting upon them
that their hypothesis of the being of the many if
carried out appears in a still more ridiculous light than
the hypothesis of the being of the one."

Zeno's four arguments against motion were intended
to exhibit the contradictions that result from supposing
that there is such a thing as change, and thus to support
the Parmenidean doctrine that reality is unchanging. 1

' This interpretation is combated by Milhaud, Les philo-

THE PROBLEM OF INFINITY 173

Unfortunately, we only know his arguments through
Aristotle, 1 who stated them in order to refute them.
Those philosophers in the present day who have had
their doctrines stated by opponents will realize that
a just or adequate presentation of Zeno's position is
hardly to be expected from Aristotle ; but by some
care in interpretation it seems possible to reconstruct
the so-called " sophisms " which have been " refuted "
by every tyro from that day to this.

Zeno's arguments would seem to be " ad hominem " ;
that is to say, they seem to assume premisses granted
by his opponents, and to show that, granting these
premisses, it is possible to deduce consequences which
his opponents must deny. In order to decide whether
they are valid arguments or " sophisms," it is necessary
to guess at the tacit premisses, and to decide who was
the " homo " at whom they were aimed. Some main-
tain that they were aimed at the Pythagoreans, 3
while others have held that they were intended to
refute the atomists.s M. Evellin, on the contrary,
holds that they constitute a refutation of infinite
divisibility^ while M. G. Noel, in the interests of
Hegel, maintains that the first two arguments refute

sophes-gtometrcs da la Greet, p. 140 n. ( but his reasons do not
seem to me convincing. All the interpretations in what
follows are open to question, but all have the support of
reputable authorities.

Physics, vi. 9. 2396 (R-P- 136-139).

* Cf . Gaston Milhaud, Les philosophes-g&m&tres de la Grece t
p. 140 n. ; Paul Tannery, Pour I'histoire de la science hellene,
p. 249 ; Buniet, op. tit., p. 362.

s Cf. R. K. Gaye, "On Aristotle, Physics, Z be." Journal
of Philology, vol. xxxi. esp. p. HI. Also Moritz Cantor,
Vorlesungen fiber Geschichte der Mathematik, ist ed. f vol. i.,
1880, p. 168, who, however, subsequently adopted Paul
Tannery's opinion, Vorlesungen, 3rd ed. (vol. i. p. 200).

4 " Le mouvement et les partisans des indivisibles," Re we
de MJtaphysique et de Morale, vol. i. pp. 382-395-

174 SCIENTIFIC METHOD IN PHILOSOPHY

infinite divisibility, while the next two refute indi-
visibles. 1 Amid such a bewildering variety of inter-
pretations, we can at least not complain of any
restrictions on our liberty of choice.

The historical questions raised by the above-men-
tioned discussions are no doubt largely insoluble, owing
to the very scanty material from which our evidence
is derived. The points which seem fairly clear axe
the following : (i) That, in spite of MM. Milhaud and
Paul Tannery, Zeno is anxious to prove that motion
is really impossible, and that he desires to prove this
because he follows Parmenides in denying plurality ; a
(2) that the third and fourth arguments proceed on
the hypothesis of indivisibles, a hypothesis which,
whether adopted by the Pythagoreans or not, was
certainly much advocated, as may be seen from the
treatise On Indivisible Lines attributed to Aristotle.
As regards the first two arguments, they would seem
to be valid on the hypothesis of indivisibles, and also,
without this hypothesis, to be such as would be valid
if the traditional contradictions in infinite numbers
were insoluble, which they are not.

We may conclude, therefore, that Zeno's polemic
is directed against the view that space and time
consist of points and instants ; and that as against the
view that a finite stretch of space of time consists of
a finite number of points and instants, his arguments
are not sophisms, but perfectly valid.

The conclusion which Zeno wishes us to draw is that
plurality is a delusion, and spaces and times are really
indivisible. The other conclusion which is possible,

' "Le mouvcinent et les arguments de Z6non d'filde,"
Revue de Mttaphysique et de Morale, vol. i. pp. 107-125.

Gf . N. Brochard, " Les prftendus sophismes de Z6non
d'filee, " Revue de MJtaphysique et de Morale, vol. i. pp. 209-215.

THE PROBLEM OF INFINITY 175

namely that the number of points and instants is
infinite, was not tenable so long as the infinite was
infected with contradictions. In a fragment which
is not one of the four famous arguments against motion,
Zeno says :

" If things are a many, they must be just as many
as they are, and neither more nor less. Now, if
they are as many as they are, they wi]l be finite in
number.

" If things are a many, they will be infinite in
number ; for there will always be other things between
them, and others again between these. And so things
are infinite in number." *

This argument attempts to prove that, if there are
many things, the number of them must be both finite
and infinite, which is impossible ; hence we are to
conclude that there is only one thing. But the weak
point in the argument is the phrase : " If they are
just as many as they axe, they will be finite in number."
This phrase is not very dear, but it is plain that it
assumes the impossibility of definite infinite numbers.
Without this assumption, which is now known to
be false, the arguments of Zeno, though they suffice
(on certain very reasonable assumptions) to dispel
the hypothesis of finite indivisibles, do not suffice to
prove that motion and change and plurality are im-
possible. They are not, however, on any view, mere
foolish quibbles : they are serious arguments, raising
difficulties which it has taken two thousand years to
answer, and which even now are fatal to the teachings
of most philosophers.

The first of Zeno's arguments is the argument of

i Simplicius, Phys., 140, 28 D (R.P. 133) ; Biirnet, op. cit. f
pp. 364-365-

176 SCIENTIFIC METHOD IN PHILOSOPHY

the race-course, which is paraphrased by Burnet as
follows : x

" You cannot get to the end of a race-course. You
cannot traverse an infinite number of points in a finite
time. You must traverse the half of any given dis-
tance before you traverse the whole, and the half of
that again before you can traverse it. This goes on
ad infinitum, so that there are an infinite number of
points in any given space, and you cannot touch an
infinite number one by one in a finite time." a

Zeno appeals here, in the first place, to the fact that
any distance, however small, can be halved. From
this it follows, of course, that there must be an infinite
number of points in a line. But Aristotle represents

' Op. cit., p. 367.

Aristotle's words are : " The first is the one on the non-
existence of motion on the ground that what is moved must
always attain, the middle point sooner than the end-point, on
which we gave our opinion in the earlier part of our discourse."
Phys., vi 9. 9398 (R.P. 136). Aristotle seems to refer to
Phys., vi. 2. 223AB [R-P. I3*>A] .: " All space is continuous,
for t and space are divided into the same and equal divisions.
. . . Wherefore also Zeno's argument is fallacious, that it is
sible to go through an infinite collection or to touch an

infinite collection one by one in a finite time. For there are
two senses in which the term ' infinite ' is applied both to
length and to time, and in fact to all continuous things, either
in regard to divisibility, or in regard to the ends. Now it is
not possible to touch things infinite in regard to number in a
finite time, but it is possible to touch things infinite in regard
to divisibility: for time itself also is infinite in this sense.
So that in fact we go through an infinite [space], in an infinite
[time] and not in a finite [time], and we touch infinite things
with infinite things, not with finite things." Philoponus, a
sixth-century commentator (R.P. I36A, Ex&. Paris Philop. in
Arist. Phys., 803, 2. Vit.), gives the following illustration:
" For if a thing were moved the space of a cubit in one hour,
since in every space there are an infinite number of points,
the thing moved must needs touch all the points of the space :
it will then go through an infinite collection in a finite time,
which is impossible."

THE PROBLEM OF INFINITY 177

him as arguing, you cannot touch an infinite number of
points one by one in a finite time. The words " one
by one " are important, (i) If att the points touched
are concerned, then, though you pass through them
continuously, you do not touch them " one by one/'
That is to say, after touching one, there is not another
which you touch next : no two points are next each
other, but between any two there are always an infinite
number of others, which cannot be enumerated one
by one. (2) If, on the other hand, only the successive
middle points are concerned, obtained by always
halving what remains of the course, then the points
are reached one by one, and, though they are infinite
in number, they are in fact all reached in a finite
time. His argument to the contrary may be supposed
to appeal to the view that a finite time must consist
of a finite number of instants, in which case what he
says would be perfectly true on the assumption that
the possibility of continued dichotomy is undeniable.
If, on the other hand, we suppose the argument
directed against the partisans of infinite divisibility,
we must suppose it to proceed as follows : x " The
points given by successive halving of the distances
still to be traversed are infinite in number, and are
reached in succession, each being reached a finite
time later than its predecessor ; but the sum of an
infinite number of finite times must be infinite, and
therefore the process will never be completed/* It is
very possible that this is historically the right inter-
pretation, but in this form the argument is invalid.
If half the course takes half a minute, and the next
quarter takes a quarter of a minute, and so on, the
whole course will take a minute. The apparent

' a. Mr. C. D. Broad, " Note on Achilles and the Tortoise/
Mind. N.S,, vol. xxii. pp. 3iS-g.

12

178 SCIENTIFIC METHOD IN PHILOSOPHY

force of the argument, on this interpretation, lies
solely in the mistaken supposition that there cannot
be anything between the whole of an infinite series,
which can be seen to be false by observing that i is
beyond the whole of the infinite series, J, f, f , -J-$-, . . .

The second of Zeno's arguments is the one concern-
ing Achilles and the tortoise, which has achieved more
notoriety than the others. It is paraphrased by
Burnet as follows : I

"Achilles will never overtake the tortoise. He
must first reach the place from which the tortoise
started. By that time the tortoise will have got
some way ahead. Achilles must then make up
that, and again the tortoise will be ahead. He
is always coming nearer, but he never makes up
to it." *

This argument is essentially the same as the previous
one. It shows that, if Achilles ever overtakes the
tortoise, it must be after an infinite number of instants
have elapsed since he started. This is in fact true ;
but the view that an infinite number of instants make
up an infinitely long time is not true, and therefore
the conclusion that Achilles will never overtake the
tortoise does not follow.

The third argument^ that of the arrow, is very
interesting. The text has been questioned. Burnet
accepts the alterations of Zeller, and paraphrases
thus:

* Op. cit.

Aristotle's words are: "The second is the so-called
Achilles. It consists in this, that the slower will never be
overtaken in its course by the quickest, for the pursuer must
always come first to the point from which the pursued has
just departed, so that the slower must necessarily be always
still more or less in advance." Phys., vi. 9. 2393 (R.P. 137).

I Phys., vi. 9. 2398 (R.P. 138).

THE PROBLEM OF INFINITY 179

"The arrow in flight is at rest. For, if every-
thing is at rest when it occupies a space equal to
itself, and what is in flight at any given moment
always occupies a space equal to itself, it cannot
move."

But according to Prantl, the literal translation
of the unemended text of Aristotle's statement of the
argument is as follows : " If everything, when it is
behaving in a uniform manner, is continually either
moving or at rest, but what is moving is always in
the now, then the moving arrow is motionless." This
form of the argument brings out its force more clearly
than Bumet's paraphrase.

Here, if not in the first two arguments, the view that
a finite part of time consists of a finite series of suc-
cessive instants seems to be assumed ; at any rate
the plausibility of the argument seems to depend upon
supposing that there are consecutive instants.
Throughout an instant, it is said, a moving body is
where it is : it cannot move during the instant, for
that would require that the instant should have parts.
Thus, suppose we consider a period consisting of a
thousand instants, and suppose the arrow is in flight
throughout this period. At each of the thousand
instants, the arrow is where it is, though at the next
instant it is somewhere else. It is never moving,
but in some miraculous way the change of position
has to occur between the instants, that is to say, not
at any time whatever. This is what M. Bergson calls
the cinematographic representation of reality. The
more the difficulty is meditated, the more real it
becomes. The solution lies in the theory of continuous
series : we find it hard to avoid supposing that, when
the arrow is in flight, there is a next position occupied
at the next moment ; but in fact there is no next

i8o SCIENTIFIC METHOD IN PHILOSOPHY

position and no next moment, and when once
this is imaginatively realized, the difficulty is seen to
disappear.

The fourth and last of Zeno's arguments is x the
argument of the stadium.

The argument as stated by Burnet is as follows :

First Position. Second Position.

A A

" Half the time may be equal to double the time.
Let us suppose three rows of bodies, one of which
(A) is at rest while the other two (B, C) are moving
with equal velocity in opposite directions. By the
time they axe all in the same part of the course, B
will have passed twice as many of the bodies in C as
in A. Therefore the time which it takes to pass C
is twice as long as the time it takes to pass A. But
the time which B and C take to reach the position of
A is the same. Therefore double the time is equal
to the hajf."

Gayety devoted an interesting article to the inter-
pretation of this argument. His translation of Aris-
totle's statement is as follows :

" The fourth argument is that concerning the two
rows of bodies, each row being composed of an equal
number of bodies of equal size, passing each other on
a race-course as they proceed with equal velocity
in opposite directions, the one row originally occupying
the space between the goal and the middle point of
the course, and the other that between the middle

' Phys., vi. 9. 2393 (RJP. 139).
Loc. cit.

THE PROBLEM OF INFINITY 181

point and the starting-post. This, he thinks, involves
the conclusion that half a given time is equal to double
the time. The fallacy of the reasoning lies in the
assumption that a body occupies an equal time in
passing with equal velocity a body that is in motion and
a body of equal size that is at rest, an assumption which
is false. For instance (so runs the argument), let
A A . . . be the stationary bodies of equal size,
BB . . . the bodies, equal in number and in size
to A A . . ., originally occupying the half of the course
from the starting-post to the middle of the A's, and
CC . . . those originally occupying the other half from
the goal to the middle of the A's, equal in number, size,
and velocity, to BB . .'. Then three consequences
follow. First, as the B's and C's pass one another, the
first B reaches the last C at the same moment at
which the first C reaches the last B. Secondly, at
this moment the first C has passed all the A's, whereas
the first B has passed only half the A's and has conse-
quently occupied only half the time occupied by the
first C, since each of the two occupies an equal time
in passing each A. Thirdly, at the same moment
all the B's have passed all the C's : for the first C and
the first B will simultaneously rea^ht the opposite
ends of the course, since (so says Zeno) the time occupied
by the first C in passing each of the B's is equal to
that occupied by it in passing each of the A's, be-
cause an equal time is occupied by both the first B
and the first C in passing all the A's. This is the
argument : but it presupposes the aforesaid fallacious
assumption."

This argument is not quite easy to follow, and
it is only valid as against the assumption that
a finite time consists of a finite number of instants.
We may re-state it in different language. Let us

i82 SCIENTIFIC METHOD IN PHILOSOPHY

suppose three drill-sergeants, A, A', and A", standing in
a row, while the two files of soldiers march past them in

First Position. Second Position.

B B' B" B B' B"

A A' A" A A' A"

C C' C" C C' C"

opposite directions. At the first moment which we
consider, the three men B, B', B" in one row, and the
three men C, C', C" in the other row, are respectively
opposite to A, A', and A". At the very next moment,
each row has moved on, and now B and C" are opposite
A'. Thus'B and C" are opposite each other. When,
then, did B pass C' ? It must have been somewhere
between the two moments which we supposed con-
secutive, and therefore the two moments cannot really
have been consecutive. It follows that there must
be other moments between any two given moments,
and therefore that there must be an infinite number of
moments in any given interval of time.

The above difficulty, that B must have passed C'
at some time between two consecutive moments, is a
genuine one, but is not precisely the difficjjlty raised by
Zeno. What Zeno professes to prove tethat," half of
a given time is equal to double that time." The most
intelligible explanation of the argument known to me
is that of Gaye. 1 Since, however, his explanation is
not easy to set forth shortly /I will re-state what
seems to me to be the logical essence of Zeno's conten-
tion. If we suppose that time consists of a series of

1 Loc. tit., p. 105.

THE PROBLEM OF INFINITY 183

consecutive instants, and that motion consists in
passing through a series of consecutive points, then
the fastest possible motion is one which, at each instant,
is at a point consecutive to that at which it was at the
previous instant. Any slower motion must be one
which has intervals of rest interspersed, and any
faster motion must wholly omit some points. All this
is evident from the fact that we cannot have more than
one event for each instanj. But now, in the case of
our A's and B's and C's, B is opposite a fresh A every
instant, and therefore the number of A's passed gives
the number of instants since the beginning of the
motion. But during the motion B has passed twice as
many C's, and yet cannot have passed more than one
each instant. Hence the number of instants since the
motion began is twice the number of A's passed,
though we previously found it was equal to this number.
From this result, Zeno's conclusion follows.

Zeno's arguments, in some form, have afforded
grounds for almost all the theories of space and time
and infinity which have been constructed from his
day to our own. We have seen that all his arguments
are valid (with certain reasonable hypotheses) on the
assumption that finite spaces and times consist of a
finite number of points and instants, and that the
third and fourth almost certainly in fact proceeded
on this assumption, while the first and second, which
were perhaps intended to refute the opppsite assump-
tion, were in that case fallacious. We may therefore
escape from his paradoxes either by maintaining
that, though space and time do consist of points and
instants, the number of them in any finite interval is
infinite ; or by denying that space and time consist
of points and instants at all ; or lastly, by denying
the reality of space and time altogether. It would

184 SCIENTIFIC METHOD IN PHILOSOPHY

seem that Zeno himself, as a supporter of Parmenides,
drew the last of these three possible deductions, at
any rate in regard to time. In this a very large
number of philosophers have followed him. Many
others, like M. Bergson, have preferred to deny that
space and time consist of points and instants. Either
of these solutions will meet the difficulties in the form
in which Zeno raised them. But, as we saw, the diffi-
culties can also be met if infinite numbers are admis-
sible. And on grounds which are independent of
space and time, infinite numbers, and series in which
no two terms are consecutive, must in any case be
admitted. Consider, for example, all the fractions
less than I, arranged in order of magnitude. Between
any two of them, there are others, for example, the
arithmetical mean of the two. Thus no two fractions
are consecutive, and the total number of them is
infinite. It will be found that much of what Zeno says
as regards the series of points on a line can be equally
well applied to the series of fractions. And we cannot
deny that there are fractions, so that two of the above
ways of escape are closed to us. It follows that, if
we are to solve the whole class of difficulties derivable
from Zeno's by Analogy, we must discover some tenable
theory of infinite numbers. What, then, are the
difficulties which, until the last thirty years, led
philosophers to the belief that infinite numbers are
impossible ?

The difficulties of infinity are of two kinds, of which
the first may be called sham, while the others involve,
for their solution, a certain amount of new and not
altogether easy thinking. The sham difficulties are
those suggested by the etymology, and those suggested
by confusion of the mathematical infinite with what
philosophers impertinently call the "true" infinite. 1

THE PROBLEM OF INFINITY 185

Etymologically, "infinite" should mean "having
no end." But in fact some infinite series have ends,
some have not; while some collections are infinite
without being serial, and can therefore not properly
be regarded as either endless or having ends. The
series of instants from any earlier one to any later one
(both included) is infinite, but has two ends; the
series of instants from the beginning of time to the
present moment has one end, but is infinite. Kant,
in his first antinomy, seems to hold that it is harder
for the past to be infinite than for the future to be so,
on the ground that the past is now completed, and
that nothing infinite can be completed. It is very
difficult to see how he can have imagined that there
was any sense in this remark; but it seems most
probable that he was thinking of the infinite as the
" unended." It is odd that he did not see that the
future too has one end at the present, and is precisely
on a level with the past. His regarding the two as
different in this respect illustrates just that kind of
slavery to time which, as we agreed in speaking of
Pannenides, the true philosopher must learn to leave
behind him.

The confusions introduced into the notions of philo-
sophers by the so-called " true " infinite are curious.
They see that this notion is not the same as the mathe-
matical infinite, but they choose to believe that it is
the notion which the mathematicians are vainly
trying to reach. They therefore inform the mathe-
maticians, kindly but firmly, that they are mistaken
in adhering to the " false " infinite, since plainly the
"true" infinite is something quite different. The
reply to this is that what they call the " true " infinite
is a notion totally irrelevant to the problem of the
mathematical infinite, to which it has only a fanciful

i86 SCIENTIFIC METHOD IN PHILOSOPHY

and verbal analogy. So remote is it that I do not
propose to confuse the issue by even mentioning what
the " true " infinite is. It is the " false " infinite that
concerns us, and we have to show that the epithet
" false " is undeserved.

There are, however, certain genuine difficulties in
understanding the infinite, certain habits of mind
derived from the consideration of finite numbers, and
easily extended to infinite numbers under the mistaken
notion that they represent logical necessities. For
example, every number that we are accustomed to,
except o, has another number immediately before
it, from which it results by adding i ; but the first
infinite number does not have this property. The
numbers before it form an infinite series, containing
all the ordinary finite numbers, having no maximum,
no last finite number, after which one little step would
plunge us into the infinite. , If it is assumed that the
first infinite number is reached by a succession of
small steps, it is easy to show that it is self-contra-
dictory. The first infinite number is, in fact, beyond
the whole unending series of finite numbers. " But,"
it will be said, "there cannot be anything beyond
the whole of an unending series." This, we may
point out, is the very principle upon which Zeno relies
in the arguments of the race-course and the Achilles.
Take the race-course : there is the moment when the
runner still has half his distance to run, then the
moment when he still has a quarter, then when he still
has an eighth, and so on in a strictly unending series.
Beyond the whole of this series is the moment when
he reaches the goal Thus there certainly can be
something beyond the whole of an unending series.
But it remains to show that this fact is only what
might have been expected.

THE PROBLEM OF INFINITY 187

The difficulty, like most of the vaguer difficulties
besetting the mathematical infinite, is derived, I
think, from the more or less unconscious operation
of the idea of counting. If you set to work to count
the terms in an infinite collection, you will never
have completed your task. Thus, in the case of the
runner, if half, three-quarters, seven-eighths, and so
on of the course were marked, and the runner was not
allowed to pass any of the marks until the umpire
said "Now," then Zeno's conclusion would be true
in practice, and he would never reach the goal.

But it is not essential to the existence of a collection,
or even to knowledge and reasoning concerning it,
that we should be able to pass its terms in review one
by one. This may be seen in the case of finite col-
lections ; we can speak of " mankind " or " the human
race," though many of the individuals in this collection
are not personally known to us. We can do this
because we know of various characteristics which every
individual has if he belongs to the collection, and not
if he does not. And exactly the same happens in the
case of infinite collections : they may be known by
their characteristics although their terms cannot be
enumerated. In this sense, an unending series may
nevertheless form a whole, and there may be new
terms beyond the whole of it.

Some purely arithmetical peculiarities of infinite
numbers have also caused perplexity. For instance,
an infinite number is not increased by adding one to
it, or by doubling it. Such peculiarities have seemed
to many to contradict logic, but in fact they only
contradict confirmed mental habits. The whole diffi-
culty of the subject lies in the necessity of thinking
in an nnfa-miliflr way, and in realizing that many
properties which we have thought inherent in number

i88 SCIENTIFIC METHOD IN PHILOSOPHY

are in fact peculiar to finite numbers. If this is
remembered, the positive theory of infinity, which
will occupy the next lecture, will not be found so
difficult as it is to those who ding obstinately to the
prejudices instilled by the arithmetic which is learnt
in childhood.

LECTURE VH
THE POSITIVE THEORY OF INFINITY

THE positive theory of infinity, and the general theory
of number to which it has given rise, are among the
triumphs of scientific method in philosophy, and are
therefore specially suitable for illustrating the logical-
analytic character of that method. The work in this
subject has been done by mathematicians, and its
results can be expressed in mathematical symbolism.
Why, then, it may be said, should the subject be
regarded as philosophy rather than as mathematics ?
This raises a difficult question, partly concerned with
the use of words, but partly also of real importance in
understanding the function of philosophy. Every
subject-matter, it would seem, can give rise to philo-
sophical investigations as well as to the appropriate
science, the difference between the two treatments
being in the direction of movement and in the kind of
truths which it is sought to establish. In the special
sciences, when they have become fully developed, the
movement is forward and synthetic, from the simpler
to the more complex. But in philosophy we follow
the inverse direction : from the complex and relatively
concrete we proceed towards the simple and abstract
by means of analysis, seeking, in the process, to
eliminate the particularity of the original subject-

igo SCIENTIFIC METHOD IN PHILOSOPHY

matter, and to confine our attention entirely to the
logical form of the facts concerned.

Between philosophy and pure mathematics there is
a certain affinity, in the fact that both are general
and a priori. Neither of them asserts propositions
which, like those of history and geography, depend
upon the actual concrete facts being just what they
are. We may illustrate this characteristic by means
of Leibniz's conception of many possible worlds, of
which one only is actual. In all the many possible
worlds, philosophy and mathematics will be the
same ; the differences will only be in respect of those
particular facts which are chronicled by the descriptive
sciences. Any quality, therefore, by which our actual
world is distinguished from other abstractly possible
worlds, must be ignored by mathematics and philo-
sophy alike. Mathematics and philosophy differ,
however, in their manner of treating the general
properties in which all possible worlds agree ; for
while mathematics, starting from comparatively simple
propositions, seeks to build up more and more complex
results by deductive synthesis, philosophy, starting
from data which are common knowledge, seeks to
purify and generalize them into the simplest statements
of abstract form that can be obtained from them by
logical analysis.

The difference between philosophy and mathematics
may be illustrated by our present problem, namely
the nature of number. Both start from certain facts
about numbers which are evident to inspection. But
mathematics uses these facts to deduce more and
more complicated theorems, while philosophy seeks,
by analysis, to go behind tEese facts to others, simpler,
more fundamental, and inherently more fitted to form
the premisses of the science of arithmetic. The

THE POSITIVE THEORY OF INFINITY 191

question, " What is a number ? " is the pre-eminent
philosophic question in this subject, but it is one which
the mathematician as such need not ask, provided he
knows enough of the properties of numbers to enable
him to deduce his theorems. We, since our object is
philosophical, must grapple with the philosopher's
question. The answer to the question, " What is a
number ? " which we shall reach in this lecture, will
be found to give also, by implication, the answer to
the difficulties of infinity which we considered in the
previous lecture.

The question " What is a number ? " is one which,
until quite recent times, was never considered in the
kind of way that is capable of yielding a precise answer.
Philosophers were content with some vague dictum
such as " Number is unity in plurality." A typical
definition of the kind that contented philosophers is
the following from Sigwart's Logic ( 66, section 3) :
" Every number is not merely a plurality, but a plur-
ality thought as held together and dosed, and to that
extent as a unity' 9 Now there is in such definitions a
very elementary blunder, of the same kind that would
be committed if we said " yellow is a flower " because
some flowers are yellow. Take, for example, the
number 3. A single collection of three things might
conceivably be described as " a plurality thought as
held together and closed, and to that extent as a
unity " ; but a collection of three things is not the
number 3. The number 3 is something which all
collections of three things have in common, but is not
itself a collection of three things. The definition,
therefore, apart from any other defects, has failed to
reach the necessary degree of abstraction : the number
3 is something -more abstract than any collection of
three things.

IQ2 SCIENTIFIC METHOD IN PHILOSOPHY

Such vague philosophic definitions, however, re-
mained inoperative because of their very vagueness.
What most men who thought about numbers really
had in mind was that numbers are the result of counting.
" On the consciousness of the law of counting/' says
Sigwart at the beginning of his discussion of number,
" rests the possibility of spontaneously prolonging the
series of numbers ad infinitum." It is this view of
number as generated by counting which has been the
chief psychological obstacle to the understanding of
infinite numbers. Counting, because it is familiar,
is erroneously supposed to be simple, whereas it is in
fact a highly complex process, which has no meaning
unless the numbers reached in counting have some
significance independent of the process by which they
axe reached. And infinite numbers cannot be reached
at all in this way. The mistake is of the same kind
as if cows were defined as what can be bought from a
cattle-merchant. To a person who knew several cattle-
merchants, but had never seen a cow, this might seem
an admirable definition. But if in his travels he came
across a herd of wild cows, he would have to declare
that they were not cows at all, because no cattle-
merchant could sell them. So infinite numbers were
declared not to be numbers at all, because they could
not be reached by counting.

It will be worth while to consider for a moment what
counting actually is. We count a set of objects when
we let our attention pass from one to another, until we
have attended once to each, saying the names of the
numbers in order with each successive act of atten-
tion. The last number named in this process is the
number of the objects, and therefore counting is a
method of finding out what the number of the objects
is. But this operation is really a very complicated

THE POSITIVE THEORY OF INFINITY 193

one, and those who imagine that it is the logical
source of number show themselves remarkably in-
capable of analysis. In the first place, when we say
" one, two, three ..." as we count, we cannot be
said to be discovering the number of the objects
counted unless we attach some meaning to the words
one, two, three. ... A child may learn to know
these words in order, and to repeat them correctly like
the letters of the alphabet, without attaching any
meaning to them. Such a child may count correctly
from the point of view of a grown-up listener, without
having any idea of numbers at all. The operation of
counting, in fact, can only be intelligently performed
by a person who already has some idea what the
numbers are ; and from this it follows that counting
does not give the logical basis of number.

Again, how do we know that the last number reached
in the process of counting is the number of the objects
counted ? This is just one of those facts that are too
familiar for their significance to be realized; but
those who wish to be logicians must acquire the habit
of dwelling upon such facts. There are two proposi-
tions involved in this fact : first, that the number of
numbers from i up to any given number is that given
number for instance, the number of numbers from
i to 100 is a hundred ; secondly, that if a set of numbers
can be used as names of a set of objects, each number
occurring only once, then the number of numbers
used as names is the same as the number of objects.
The first of these propositions is capable of an easy
arithmetical proof so long as finite numbers are con-
cerned ; but with infinite numbers, after the first, it
ceases to be true. The second proposition remains
true, and is in fact, as we shall see, an immediate
consequence of the definition of number. But owing

13

194 SCIENTIFIC METHOD IN PHILOSOPHY

to the falsehood of the first proposition where infinite
numbers are concerned, counting, even if it were
practically possible, would not be a valid method of
discovering the number of terms in an infinite collec-
tion, and would in fact give different results according
to the manner in which it was carried out.

There are two respects in which the infinite numbers
that are known differ from finite numbers : first,
infinite numbers have, while finite numbers have not,
a property which I shall call reflexiveness ; secondly,
finite numbers have, while infinite numbers have not,
a property which I shall call inductiveness. Let us
consider these two properties successively.

(i) Reflexiveness. A number is said to be reflexive
when it is not increased by adding i to it. It follows
at once that any finite number can be added to a
reflexive number without increasing it. This property
of infinite numbers was always thought, until recently,
to be self-contradictory; but through the work of
Georg Cantor it has come to be recognized that, though
at first astonishing, it is no more self-contradictory
than the fact that people at the antipodes do not
tumble off. In virtue of this property, given any
infinite collection of objects, any finite number of
objects can be added or taken away without increasing
or diminishing the number of the collection. Even
an infinite number of objects may, under certain
conditions, be added or taken away without altering
the number. This may be made clearer by the help
of some examples.

Imagine all the natural numbers o, i, 2, 3 ... to
be written down in a row, and immediately beneath
them

O, I, 2, 3 ^

i* 2, 3* 4> * r i

THE POSITIVE THEORY OF INFINITY 195

write down the numbers i, 2, 3, 4, . . ., so that i is
under o, 2 is under i, and so on. Then every number
in the top row has a number directly under it in the
bottom row, and no number occurs twice in either
row. It follows that the number of numbers in the
two rows must be the same. But all the numbers
that occur in the bottom row also occur in the top
row, and one more, namely o ; thus the number of
terras in the top row is obtained by adding one to
the number of the bottom row. So long, therefore,
as it was supposed that a number must be increased
by adding i to it, this state of things constituted a
contradiction, and led to the denial that there are
infinite numbers.

The following example is even more surprising.
Write the natural numbers i, 2, 3, 4 ... in the top
row, and the even numbers 2, 4, 6, 8 ... in the
bottom row, so that under each number in the top
row stands its double in the bottom row. Then, as
before, the number of numbers in the two rows is the
same, yet the second row results from taking away
all the odd numbers an infinite collection from the
top row. This example is given by Leibniz to prove
that there can be no infinite numbers. He believed
in infinite collections, but, since he thought that a
number must always be increased when it is added to
and diminished when it is subtracted from, he main-
tained that infinite collections do not have numbers.
" The number of all numbers," he says, " implies a
contradiction, which I show thus: To any number
there is a corresponding number equal to its double.
Therefore the number of all numbers is not greater
tfcm the number of even numbers, i.e. the whole is
not greater thft" its part." x In dealing with this
Phil. Werke, Gerhardt's edition, vol. i. p. 338.

196 SCIENTIFIC METHOD IN PHILOSOPHY

argument, we ought to substitute " the number of all
finite numbers " for " the number of all numbers " ;
we then obtain exactly the illustration given by our
two rows, one containing all the finite numbers, the
other only the even finite numbers. It will be seen
that Leibniz regards it as self-contradictory to main-
tain that the whole is not greater than its part. But
the word " greater " is one which is capable of many
meanings ; for our purpose, we must substitute the
less ambiguous phrase " containing a greater number
of terms." In this sense, it is not self-contradictory
for whole and part to be equal ; it is the realization
of this fact which has made the modern theory of
infinity possible.

There is an interesting discussion of the reflexiveness
of infinite wholes in the first of Galileo's Dialogues on
Motion. I quote from a translation published in
I730. 1 The personages in the dialogue are Salviati,
Sagredo, and Simplicius, and they reason as follows :

" Simp. Here already arises a Doubt which I
think is not to be resolv'd ; and that is this : Since
'tis plain that one Line is given greater than another,
and since both contain infinite Points, we must surely
necessarily infer, that we have found in the same
Species something greater than Infinite, since the
Infinity of Points of the greater Line exceeds the
Infinity of Points of the lesser. But now, to assign
an Infinite greater than an Infinite, is what I can't
possibly conceive.

i Mathematical Discourses concerning two new sciences
relating to mechanics and local motion t in four dialogues. By
Galileo Galilei, Chief Philosopher and Mathematician to the
Grand Duke of Tuscany. Done into English from the Italian,
by Tho. Weston, late Master, and now published by John
Weston, present Master, of the Academy at Greenwich.
See pp. 46 ff.

THE POSITIVE THEORY OF INFINITY 197

" Salv. These are some of those Difficulties which
arise from Discourses which our finite Understanding
makes about Infinites, by ascribing to them Attributes
which we give to Things finite and terminate, which I
think most improper, because those Attributes of
Majority, Minority, and Equality, agree not with
Infinities, of which we can't say that one is greater
than, less than, or equal to another. For Proof
whereof I have something come into my Head, which
(that I may be the better understood) I will propose
by way of Interrogatories to Simplicity, who started
this Difficulty. To begin then : I suppose you know
which are square Numbers, and which not ?

" Simp. I know very well that a square Number is
that which arises from the Multiplication of any
Number into itself ; thus 4 and 9 are square Numbers,
that arising from 2, and this from 3, multiplied by
themselves.

"Salv. Very well; And you also know, that as
the Products are calTd Squares, the Factors are calTd
Roots : And that the other Numbers, which proceed
not from Numbers multiplied into themselves, are not
Squares. Whence taking in all Numbers, both Squares
and Not Squares, if I should say, that the Not Squares
are more than the Squares, should I not be in the
right?

" Simp. Most certainly.

" Salv. If I go on with you then, and ask you, How
many squar'd Numbers there are? you may truly
answer, That there are as many as are their proper
Roots, since every Square has its own Root, and
every Root its own Square, and since no Square has
more than one Root, nor any Root more than one
Square.

" Simp. Very true.

IQ8 SCIENTIFIC METHOD IN PHILOSOPHY

But now, if I should ask how many Roots
there are, you can't deny but there are as many as
there are Numbers, since there's no Number but
what's the Root to some Square. And this being
granted, we may likewise affirm, that there are as many
square Numbers, as there are Numbers ; for there are
as many Squares as there are Roots, and as many
Roots as Numbers. And yet in the Beginning of this,
we said, there were many more Numbers than Squares,
the greater Part of Numbers being not Squares : And
tho' the Number of Squares decreases in a greater
proportion, as we go on to bigger Numbers, for count
to an Hundred you'll find 10 Squares, viz. i, 4, 9, 16,
25, 36, 49, 64, 81, 100, which is the same as to say the
loth Part are Squares; in Ten thousand only the
xooth Part are Squares ; in a Million only the loooth :
And yet in an infinite Number, if we can but compre-
hend it, we may say the Squares are as many as all
the Numbers taken together.

" Sagr. What must be determin'd then in this
Case?

" Salv. I see no other way, but by saying that all
Numbers are infinite ; Squares are Infinite, their Roots
Infinite, and that the Number of Squares is not less
than the Number of Numbers, nor this less than
that : and then by concluding that the Attributes
or Terms of Equality, Majority, and Minority, have
no Place in Infinites, but are confin'd to terminate
Quantities."

The way in which the problem is expounded in the
above discussion is worthy of Galileo, but the solution
suggested is not the right one. It is actually the case
that the number of square (finite) numbers is the same
as the number of (finite) numbers. The fact that, so
long as we confine ourselves to numbers less than

THE POSITIVE THEORY OF INFINITY 199

some given finite number, the proportion of squares
tends towards zero as the given finite number increases,
does not contradict the fact that the number of all
finite squares is the same as the number of all finite
numbers. This is only an instance of the fact, now
familiar to mathematicians, that the limit of a function
as the variable approaches a given point may not be
the same as its value when the variable actually reaches
the given point. But although the infinite numbers
which Galileo discusses are equal, Cantor has shown
that what Simplicius could not conceive is true,
namely that there are an infinite number of different
infinite numbers, and that the conception of greater
and less can be perfectly well applied to them. The
whole of Simplicius's difficulty comes, as is evident,
from his belief that, if greater and less can be applied,
a part of an infinite collection must have fewer terms
than the whole ; and when this is denied, all contra-
dictions disappear. As regards greater and less
lengths of lines, which is the problem from which the
above discussion starts, that involves a meaning of
greater and less which is not arithmetical The number
of points is the same in a long line and in a short one,
being in fact the same as the number of points in all
space. The greater and less of metrical geometry
involves the new metrical conception of congruence,
which cannot be developed out of arithmetical con-
siderations alone. But this question has not the
fundamental importance which belongs to the arith-
metical theory of infinity.

(2) Non-inductiveness.The second property by
which infinite numbers are distinguished from finite
numbers is the property of non-inductiveness. This
will be best explained by defining the positive property
of inductiveness which characterizes the finite numbers,

200 SCIENTIFIC METHOD IN PHILOSOPHY

and which is named after the method of proof known
as " mathematical induction."

Let us first consider what is meant by calling a
property " hereditary " in a given series. Take such
a property as being named Jones. If a man is named
Jones, so is his son ; we will therefore call the property
of being called Jones hereditary with respect to the
relation of father and son. If a man is called Jones,
all his descendants in the direct male line are called
Jones ; this follows from the fact that the property is
hereditary. Now, instead of the relation of father and
son, consider the relation of a finite number to its
immediate successor, that is, the relation which holds
between o and i, between I and 2, between 2 and 3,
and so on. If a property of numbers is hereditary
with respect to this relation, then if it belongs to (say)
100, it must belong also to all finite numbers greater
than 100 ; for, being hereditary, it belongs to 101
because it belongs to 100, and it belongs to 102 because
it belongs to 101, and so on where the " and so
on " will take us, sooner or later, to any finite number
greater than 100. Thus, for example, the property
of being greater than 99 is hereditary in the series of
finite numbers ; and generally, a property is hereditary
in this series when, given any number that possesses
the property, the next number must always also
possess it.

It will be seen that a hereditary property, though
it must belong to all the finite numbers greater than
a given number possessing the property, need not
belong to all the numbers less than this number. For
example, the hereditary property of being greater
than 99 belongs to 100 and all greater numbers, but
not to any smaller number. Similarly, the hereditary
property of being called Jones belongs to all the

THE POSITIVE THEORY OF INFINITY 201

descendants (in the direct male line) of those who have
this property, but not to all their ancestors, because
we reach at last a first Jones, before whom the ancestors
have no surname. It is obvious, however, that any
hereditary property possessed by Adam must belong
to all men; and similarly any hereditary property
possessed by o must belong to all finite numbers. This
is the principle of what is called "mathematical
induction/' It frequently happens, when we wish to
prove that all finite numbers have some property,
that we have first to prove that o has the property,
and then that the property is hereditary, i.e. that, if it
belongs to a given number, then it belongs to the next
number. Owing to the fact that such proofs are called
" inductive," I shall call the properties to which they
are applicable 1 "inductive" properties. Thus an
inductive property of numbers is one which is
hereditary and belongs to o.

Taking any one of the natural numbers, say 29, it is
easy to see that it must have all inductive properties.
For since such properties belong to o and are hereditary,
they belong to I ; therefore, since they are hereditary,
they belong to '2, and so on ; by twenty-nine repetitions
of such arguments we show that they belong to 29. We
may define the " inductive " numbers as all those thai
possess att inductive properties ; they will be the same
as what are called the "natural" numbers, i.e. the
ordinary finite whole numbers. To all such numbers,
proofs by mathematical induction can be validly
applied. They are those numbers, we may loosely say,
which can be reached from o by successive additions
of i ; in other words, they are all the numbers that
can be reached by counting.

But beyond all these numbers, there are the infinite
numbers, and infinite numbers do not have all inductive

202 SCIENTIFIC METHOD IN PHILOSOPHY

properties. Such numbers, therefore, may be called
non-inductive. All those properties of numbers which
are proved by an imaginaiy step-by-step process from
one number to the next are liable to fail when we
come to infinite numbers. The first of the infinite
numbers has no immediate predecessor, because there
is no greatest finite number; thus no succession of
steps.from one number to the next will ever reach from
a finite number to an infinite one, and the step-by-step
method of proof fails. This is another reason for the
supposed self contradictions of infinite number. Many
of the most familiar properties of numbers, which
custom had led people to regard as logically necessary,
are in fact only demonstrable by the step-by-step
method, and fail to be true of infinite numbers. But
so soon as we. realize the necessity of proving such
properties by mathematical induction, and the strictly
limited scope of this method of proof, the supposed
contradictions are seen to contradict, not logic, but
only our prejudices and mental habits.

The property of being increased by the addition of
i i.e. the property of non-reflexiveness may serve to
illustrate the limitations of mathematical induction.
It is easy to prove that o is increased by the addition
of i, and that, if a given number is increased by the
addition of i, so is the next number, i.e. the number
obtained by the addition of i. It follows that each
of the natural numbers is increased by the addition
of i. This follows generally from the general argument,
and follows for each particular case by a sufficient
number of applications of the argument. We first
prove that o is not equal to i ; then, since the property
of being increased by i is hereditary, it follows that
i is not equal to 2 ; hence it follows that 2 is not equal
to 3 ; if we wish to prove that 30,000 is not equal to

THE POSITIVE THEORY OF INFINITY 203

30,001, we can do so by repeating tM$ reasoning
30,000 times. But we cannot prove in this way that
all numbers are increased by the addition of i ; we
can only prove that this holds of the numbers attain-
able by successive additions of i starting from o.
The reflexive numbers, which lie beyond all those
attainable in this way, are as a matter of fact not
increased by the addition of i.

The two properties of reflexiveness and non-induc-
tiveness, which we have considered as characteristics
of infinite numbers, have not so far been proved to be
always found together. It is known that all reflexive
numbers are non-inductive, but it is not known that
all non-inductive numbers are reflexive. Fallacious
proofs of this proposition have been published by many
writers, including myself, but up to the present no
valid proof has been discovered. The infinite numbers
actually known, however, are all reflexive as well as non-
inductive ; thus, in mathematical practice, if not in
theory, the two properties are always associated. For
our purposes, therefore, it will be convenient to ignore
the bare possibility that there may be non-inductive
non-reflexive numbers, since all known numbers are
either inductive or reflexive.

When infinite numbers are first introduced to
people, they are apt to refuse the name of numbers to
them, because their behaviour is so different from that
of finite numbers that it seems a wilful misuse of terms
to call them numbers at all. In order to meet this
feeling, we must now turn to the logical basis of arith-
metic, and consider the logical definition of numbers.

The logical definition of numbers, though it seems
an essential support to the theory of infinite numbers,
was in fact discovered independently and by a different
man. The theory of infinite numbers that is to say,

204 SCIENTIFIC METHOD IN PHILOSOPHY

the arithmetical as opposed to the logical part of the
theory was discovered by Georg Cantor, and published
by him in I882-3- 1 The definition of number was
discovered about the same time by a man whose
great genius has not received the recognition it deserves
I mean Gottlob Frege of Jena. His first work,
Begrifssckrift, published in 1879, contained the very
important theory of hereditary properties in a series
to which I alluded in connection with inductiveness.
His definition of number is conitaned in his second
work, published in 1884, and entitled Die Grundlagen
der Arithmetik, eine logisch-mathematische Untersuchung
fiber den Begri/ der ZaU* It is with this book that
the logicaJ theory of arithmetic begins, and it will
repay us to consider Frege's analysis in some detail.

Frege begins by noting the increased desire for
logical strictness in mathematical demonstrations
which distinguishes modern mathematicians from their
predecessors, and points out that this must lead to a
critical investigation of the definition of number. He
proceeds to show the inadequacy of previous philo-
sophical theories, especially of the " synthetic a priori "
theory of Kant and the empirical theory of Mill. This
brings him to the question : What kind of object is it
that number can properly be ascribed to ? He points
out that physical things may be regarded as one or
many : for example, if a tree has a thousand leaves,
they may be taken altogether as constituting its

1 In his Grundlagen einer allgemeinen Mannichfaltigkeitslehre
and in articles in A eta Ma-thematic a, vol. ii.

* The definition of number contained in this book, and
elaborated in the Grundgesetee der Arithmetik (vol. i., 1893 ;
vol. ii., 1903), was rediscovered by me in ignorance of Frege's
work. I wish to state as emphatically as possible what
seems still often ignored that his discovery antedated mine
by eighteen years.

THE POSITIVE THEORY OF INFINITE 205

foliage, which would count as one, not as a thousand ;
and one pair of boots is the same object as two boots.
It follows that physical things are not the subjects of
which number is properly predicated; for when we
have discovered the proper subjects, the number to
be ascribed must be unambiguous. This leads to a
discussion of the very prevalent view that number is
really something psychological and subjective, a view
which Frege emphatically rejects. "Number," he
says, " is as little an object of psychology or an out-
come of psychical processes as the North Sea. . . .
The botanist wishes to state something which is just
as much a fact when he gives the number of petals in
a flower as when he gives its colour. The one depends
as little as the other upon our caprice. There is
therefore a certain similarity between number and
colour ; but this does not consist in the fact that both
are sensibly perceptible in external things, but in the
fact that both are objective " (p. 34).

" I distinguish the objective," he continues, " from
the palpable, the spatial, the actual. The earth's
axis, the centre of mass of the solar system, are objec-
tive, but I should not call them actual, like the earth
itself " (p. 35). He concludes that number is neither
spatial and physical, nor subjective, but non-sensible
and objective. This conclusion is important, since it
applies to all the subject-matter of mathematics and
logic. Most philosophers have thought that the
physical and the mental between them exhausted the
world of being. Some have argued that the objects
of mathematics were obviously not subjective, and
therefore must be physical and empirical ; others have
argued that they were obviously not physical, and
therefore must be subjective and mental Both sides
were right in what they denied, and wrong in what

206 SCIENTIFIC METHOD IN PHILOSOPHY

they asserted ; Frege has the merit of accepting both
denials, and finding a third assertion by recognizing
the world of logic, which is neither mental nor physical.

The fact is, as Frege points out, that no number, not
even i, is applicable to physical things, but only to
general terms or descriptions, such as " man," " satel-
lite of the earth," " satellite of Venus." The general
term "man" is applicable to a certain number of
objects : there are in the world so and so many men.
The unity which philosophers rightly feel to be neces-
sary for the assertion of a number is the unity of the
general term, and it is the general term which is the
proper subject of number. And this applies equally
when there is one object or none which falls under
the general term. " Satellite of the earth " is a term
only applicable to one object, namely, the moon.
But " one " is not a property of the moon itself, which
may equally well be regarded as many molecules:
it is a property of the general term " earth's satellite."
Similarly, o is a property of the general term " satellite
of Venus," because Venus has no satellite. Here at
last we have an intelligible theory of the number o.
This was impossible if numbers applied to physical
objects, because obviously no physical object could
have the number o. Thus, in seeking our definition
of number we have arrived so far at the result that
numbers are properties of general terms or general
descriptions, not of physical things or of mental
occurrences.

Instead of speaking of a general term, such as
"man," as the subject of which a number can be
asserted, we may, without making any serious change,
take the subject as the class or collection of objects
i.e. " mankind " in the above instance to which the
general term in question is applicable. Two general

THE POSITIVE THEORY OF INFINITY 207

terms, such as "man" and "featherless biped,"
which are applicable to the same collection of objects,
will obviously have the same number of instances;
thus the number depends upon the class, not upon
the selection of this or that general term to describe
it, provided several general terms can be found to
describe the same class. But some general term is
always necessary in order to describe a class. Even
when the terms axe enumerated, as " this and that
and the other," the collection is constituted by the
general property of being either this, or that, or the
other, and only so acquires the unity which enables
us to speak of it as one collection. And in the case
of an infinite dass, enumeration is impossible, so that
description by a general characteristic common and
peculiar to the members of the dass is the only possible
description. Here, as we see, the theory of number to
which Frege was led by purdy logical considerations
becomes of use in showing how infinite classes can be
amenable to number in spite of being incapable of
enumeration.

Frege next asks the question: When do two collections
have the same number of terms ? In ordinary life,
we decide this question by counting ; but counting,
as we saw, is impossible in the case of infinite collections,
and is not logically fundamental with finite collections.
We want, therefore, a different method of answering
our question. An illustration may hdp to make the
method dear. I do not know how many married
men there are in England, but I do know that the
number is the same as the number of married women.
The reason I know this is that the relation of husband
and wife relates one man to one woman and one woman
to one man. A relation of this sort is called a one-
one relation. The relation of father to son is called a

208 SCIENTIFIC METHOD IN PHILOSOPHY

one-many relation, because a man can have only one
father but may have many sons; conversely, the
relation of son to father is called a many-one relation.
But the relation of husband to wife (in Christian
countries) is called one-one, because a man cannot
have more than one wife, or a woman more than one
husband. Now, whenever there is a one-one relation
between all the terms of one collection and all the
terms of another severally, as in the case of English
husbands and English wives, the number of terms in
the one collection is the same as the number in the
other; but when there is not such a relation, the
number is different. This is the answer to the ques-
tion : When do two collections have the same number
of terms ?

We can now at last answer the question : What is
meant by the number of terms in a given collection ?
When there is a one-one relation between all the terms
of one collection and all the terms of another severally,
we shall say that the two collections are " similar."
We have just seen that two similar collections have
the same number of terms. This leads us to define
the number of a given collection as the class of all
collections that are similar to it ; that is to say, we set
up the following formal definition :

" The number of terms in a given class " is defined
as meaning " the class of all classes that are similar
to the given class."

This definition, as Frege (expressing it in slightly
different terms) showed, yields the usual arithmetical
properties of numbers. It is applicable equally to
finite and infinite numbers, and it does not require
the admission of some new and mysterious set of
metaphysical entities. It shows that it is not physical
objects, but classes or the general terms by which they

THE POSITIVE THEORY OF INFINITY 209

are defined, of which numbers can be asserted ; and
it applies to o and i without any of the difficulties
which other theories find in dealing with these two
special cases.

The above definition is sure to produce, at first sight,
a feeling of oddity, which is liable to cause a certain
dissatisfaction. It defines the number 2, for instance,
as the class of all couples, and the number 3 as the
class of all triads. This does not seem to be what we
have hitherto been meaning when we spoke of 2 and 3,
though it would be difficult to say what we had been
meaning. The answer to a feeling cannot be a logical
argument, but nevertheless the answer in this case is
not without importance. In the first place, it will be
found that when an idea which has grown familiar as
an unanalysed whole is first resolved accurately into
its component parts which is what we do when we
define it there is almost always a feeling of un-
famiUarity produced by the analysis, which tends to
cause a protest against the definition. In the second
place, it may be admitted that the definition, like all
definitions, is to a certain extent arbitrary. In the
case of the small finite numbers, such as 2 and 3, it
would be possible to frame definitions more nearly in
accordance with our unanalysed feeling of what we
mean ; but the method of such definitions would lack
uniformity, and would be found to fail sooner or later
at latest when we reached infinite numbers.

In the third place, the real desideratum about such
a definition as that of number is not that it should
represent as nearly as possible the ideas of those who
have not gone through the analysis required in order
to reach a definition, but that it should give us objects
having the requisite properties. Numbers, in fact,
must satisfy the formulae of arithmetic ; any indubit-

14

SCIENTIFIC METHOD IN PHILOSOPHY

able set of objects fulfilling this requirement may be
called numbers. So far, the simplest set known to
fulfil this requirement is the set introduced by the
above definition. In comparison with this merit, the
question whether the objects to which the definition
applies are like or unlike the vague ideas of numbers
entertained by those who cannot give a definition, is
one of very little importance. All the important
requirements are fulfilled by the above definition, and
the sense of oddity which is at first unavoidable will
be found to wear off very quickly with the growth
of familiarity.

There is, however, a certain logical doctrine which
may be thought to form an objection to the above
definition of numbers as classes of classes I mean
the doctrine that there are no such objects as classes
at all. It might be thought that this doctrine would
make havoc of a theory which reduces numbers to
classes, and of the many other theories in which we
have made use of classes. This, however, would be a
mistake : none of these theories are any the worse for
the doctrine that classes are fictions. What the
doctrine is, and why it is not destructive, I will try
briefly to explain.

On account of certain rather complicated difficulties,
culminating in definite contradictions, I was led to the
view that nothing that can be said significantly about
things, i.e. particulars, can be said significantly (i.e.
either truly or faJsely) about dasses of things. That
is to say, if, in any sentence in which a thing is men-
tioned, you substitute a class for the thing, you no
longer have a sentence that has any meaning : the
sentence is no longer either true or false, but a meaning-
less collection of words. Appearances to the contrary
can be dispelled by a moment's reflection. For

THE POSITIVE THEORY OF INFINITY 211

example, in the sentence, " Adam is fond of apples,"
you may substitute mankind, and say, " Mankind is
fond of apples." But obviously you do not mean
that there is one individual, called " mankind," which
munches apples : you mean that the separate indi-
viduals who compose mankind are each severally fond
of apples.

Now, if nothing that can be said significantly about
a thing can be said significantly about a class of things,
it follows that classes of things cannot have the same
kind of reality as things have ; for if they had, a dass
could be substituted for a thing in a proposition
predicating the kind of reality which would be common
to both. This view is really consonant to common
sense. In the third or fourth century B.C. there lived
a Chinese philosopher named Hui Tzu, who maintained
that " a bay horse and a dun cow are three ; because
taken separately they are two, and taken together
they are one : two and one make three." * The
author from whom I quote says that Hui Tzu " was
particularly fond of the quibbles which so delighted
the sophists or unsound reasoners of ancient Greece,"
and this no doubt represents the judgment of common
sense upon such arguments. Yet if collections of
things were things, his contention would be irrefrag-
able. It is only because the bay horse and the dun
cow taken together are not a new thing that we can
escape the conclusion that there are three things
wherever there are two.

When it is admitted that classes are not things, the
question arises : What do we mean by statements
which are nominally about classes ? Take such a
statement as, "The dass of people interested in

i Giles, The Civilisation of China (Home University Library),
p. 147-

212 SCIENTIFIC METHOD IN PHILOSOPHY

mathematical logic is not very numerous." Obviously
this reduces itself to, "Not very many people are
interested in mathematical logic." For the sake of
definiteness, let us substitute some particular number,
say 3, for " very many/' Then our statement is, " Not
three people are interested in mathematical logic."
This may be expressed in the form : " If x is interested
in mathematical logic, and also y is interested, and
also z is interested, then x is identical with y, or x is
identical with z 9 or y is identical with z." Here there
is no longer any reference at all to a " class." In some
such way, all statements nominally about a class can
be reduced to statements about what follows from
the hypothesis of anything's having the defining
property of the class. All that is wanted, therefore,
in order to render the verbal use of classes legitimate,
is a uniform method of interpreting propositions in
which such a use occurs, so as to obtain propositions
in which there is no longer any such use. The defini-
tion of such a method is a technical matter, which
Dr. Whitehead and I have dealt with elsewhere, and
which we need not enter into on this occasion. 1

If the theory that classes are merely symbolic is
accepted, it follows that numbers are not actual
entities, but that propositions in which numbers
verbally occur have not really any constituents corre-
sponding to numbers, but only a certain logical form
which is not a part of propositions having this form.
This is in fact the case with all the apparent objects
of logic and mathematics. Such words as or, not,
if, there is, identity, greater, plus, nothing, everything,
function, and so on, are not names of definite objects,
like " John " or " Jones," but are words which require

* Of. Principia Mathematics, 20, and Introduction,
chapter iii.

THE POSITIVE THEORY OF INFINITY 213

a context in order to have meaning. All of them are
formal, that is to say, their occurrence indicates a
certain form of proposition, not a certain constituent.
" Logical constants," in short, are not entities ; the
words expressing them are not names, and cannot
significantly be made into logical subjects except
when it is the words themselves, as opposed to their
meanings, that are being discussed. 1 This fact has a
very important bearing on all logic and philosophy,
since it shows how they differ from the special sciences.
But the questions raised are so large and so difficult
that it is impossible to pursue them further on this
occasion.

1 See Tractates Logico-Phihsophicus, by Lad-wig Wittgen-
stein (Kegan Paul, 1922).

LECTURE VIII

ON THE NOTION OF CAUSE, WITH APPLICA-
TIONS TO THE FREE-WILL PROBLEM

THE nature of philosophic analysis, as illustrated in
our previous lectures, can now be stated in general
terms. We start from a body of common knowledge,
which constitutes our data. On examination, the
data are found to be complex, rather vague, and
largely interdependent logically. By analysis we
reduce them to propositions which are as nearly as
possible simple and precise, and we arrange them in
deductive chains, in which a certain number of initial
propositions form a logical guarantee for all the rest.
These initial propositions are premisses for the body
of knowledge in question. Premisses are thus quite
different from data they are simpler, more precise,
and less infected with logical redundancy. If the
work of analysis has been performed completely, they
will be wholly free from logical redundancy, wholly
precise, and as simple as is logically compatible with
their leading to the given body of knowledge. The
discovery of these premisses belongs to philosophy ;
but the work of deducing the body of common know-
ledge from them belongs to mathematics, if " mathe-
matics " is interpreted in a somewhat liberal sense.

But besides the logical analysis of the common
knowledge which forms our data, there is the considera-

ON THE NOTION OF CAUSE 215

tion of its degree of certainty. When we have arrived
at its premisses, we may find that some of them seem
open to doubt, and we may find further that this
doubt extends to those of our original data which
depend upon these doubtful premisses. In our third
lecture, for example, we saw that the part of physics
which depends upon testimony, and thus upon the
existence of other minds than our own, does not seem
so certain as the part which depends exclusively upon
our own sense-data and the laws of logic. Similarly,
it used to be felt that the parts of geometry which
depend upon the axiom of parallels have less certainty
than the parts which are independent of this premiss.
We may say, generally, that what commonly passes
as knowledge is not all equally certain, and that,
when analysis into premisses has been effected, the
degree of certainty of any consequence of the premisses
will depend upon that of the most doubtful premiss
employed in proving this consequence. Thus analysis
into premisses serves not only a logical purpose, but
also the purpose of facilitating an estimate as to the
degree of certainty to be attached to this or that
derivative belief. In view of the fallibility of all
human beliefs, this service seems at least as important
as the purely logical services rendered by philosophical
analysis.

In the present lecture, I wish to apply the analytic
method to the notion of " cause, 11 and to illustrate
the discussion by applying it to the problem of free
will. For this purpose I shall inquire: I, what is
meant by a causal law ; II, what is the evidence that
causal laws have held hitherto ; III, what is the
evidence that they will continue to hold in the future ;
IV, how the causality which is used in science differs
from that of common sense and traditional philosophy ;

216 SCIENTIFIC METHOD IN PHILOSOPHY

V, what new light is thrown on the question of free
will by our analysis of the notion of " cause."

I. By a " causal law " I mean any general proposi-
tion in virtue of which it is possible to infer the exist-
ence of one thing or event from the existence of another
or of a number of others. If you hear thunder without
having seen lightning, you infer that there neverthe-
less was a flash, because of the general proposition,
"All thunder is preceded by lightning." When
Robinson Crusoe sees a footprint, he infers a human
being, and he might justify his inference by the general
proposition, " All marks in the ground shaped like a
human foot are subsequent to a human being's standing
where the marks are." When we see the sun set, we
expect that it will rise again the next day. When we
hear a man speaking, we infer that he has certain
thoughts. All these inferences are due to causal
laws.

A causal law, we said, allows us to infer the existence
of one thing (or event) from the existence of one or more
others. The word " thing " here is to be understood
as only applying to particulars, i.e. as excluding such
logical objects as numbers or classes or abstract
properties and relations, and including sense-data,
with whatever is logically of the same type as sense-
data. 1 In so far as a causal law is directly verifiable,
the thing inferred and the thing from which it is
inferred must both be data, though they need not
both be data at the same time. In fact, a causal law
which is being used to extend our knowledge of exist-
ence must be applied to what, at the moment, is not a
datum; it is in the possibility of such application

* Thus we are not using "thing " here in the sense of a class
of correlated "aspects/' as we did in Lecture III. Each
" aspect " will count separately in stating causal laws.

ON THE NOTION OF CAUSE 217

that the practical utility of a causal law consists.
The important point, for our present purpose, however,
is that what is inferred is a " thing," a " particular,"
an object having the kind of reality that belongs to
objects of sense, not an abstract object such as virtue
or the square root of two.

But we cannot become acquainted with a particular
except by its being actually given. Hence the par-
ticular inferred by a causal iaw must be only described
with more or less exactness ; it cannot be named until
the inference is verified. Moreover, since the causal
law is general, and capable of applying to many cases,
the given particular from which we infer must allow
the inference in virtue of some general characteristic,
not in virtue of its being just the particular that it is.
This is obvious in all our previous instances : we infer
the unperceived lightning from the thunder, not in
virtue of any peculiarity of the thunder, but in virtue
of its resemblance to other daps of thunder. Thus a
causal law must state that the existence of a thing of
a certain sort (or of a number of things of a number of
assigned sorts) implies the existence of another thing
having a relation to the first which remains invariable
so long as the first is of the kind in question.

It is to be observed that what is constant in a causal
law is not the object or objects given, nor yet the
object inferred, both of which may vary within wide
limits, but the relation between what is given and
what is inferred. The principle, " same cause, same
effect," which is sometimes said to be the principle of
causality, is much narrower in its scope than the
principle which really occurs in science; indeed, if
strictly interpreted, it has no scope at all, since the
" same " cause never recurs exactly. We shall return
to this point at a later stage of the discussion.

3i8 SCIENTIFIC METHOD IN PHILOSOPHY

The particular which is inferred may be uniquely
determined by the causal law, or may be only described
in such general terms that many different particulars
might satisfy the description. This depends upon
whether the constant relation affirmed by the causal
law is one which only one term can have to the data,
or one which many terms may have. If many terms
may have the relation in question, science will not be
satisfied until it has found some more stringent law,
which will enable us to determine the inferred things
uniquely.

Since all known things are in time, a causal law
must take account of temporal relations. It will be
part of the causal law to state a relation of succession
or coexistence between the thing given and the thing
inferred. When we hear thunder and infer that there
was lightning, the law states that the thing inferred
is earlier than the thing given. Conversely, when we
see lightning and wait expectantly for the thunder,
the law states that the thing given is earlier than the
thing inferred. When we infer a man's thoughts
from his words, the law states that the two are (at
least approximately) simultaneous.

If a causal law is to achieve the precision at which
science aims, it must not be content with a vague
earlier or later, but must state how much earlier or
how much later. That is to say, the time-relation
between the thing given and the thing inferred ought
to be capable of exact statement; and usually the
inference to be drawn is different according to the
length and direction of the interval. " A quarter of
an hour ago this man was alive ; an hour hence he
will be cold." Such a statement involves two causal
laws, one inferring from a datum something which
existed a quarter of an hour ago, the other inferring

ON THE NOTION OF CAUSE 219

from the same datum something which will exist an
hour hence.

Often a causal law involved not one datum, but
many, which need not be all simultaneous with each
other, though their time-relations must be given.
The general scheme of a causal law will be as
follows :

" Whenever things occur in certain relations to each
other (among which their time-relations must be
included), then a thing having a fixed relation to these
things will occur at a date fixed relatively to their
dates."

The things given will not, in practice, be things that
only exist for an instant, for such things, if there are
any, can never be data. The things given will each
occupy some finite time. They may be not static
things, but processes, especially motions. We have
considered in an earlier lecture the sense in which a
motion may be a datum, and need not now recur to
this topic.

It is not essential to a causal law that the object
inferred should be later than some or all of the data.
It may equally well be earlier or at the same time.
The only thing essential is that the law should be such
as to enable us to infer the existence of an object which
we can more or less accurately describe in terms of
the data.

II. I come now to our second question, namely :
What is the nature of the evidence that causal laws
have held hitherto, at least in the observed portions
of the past ? This question must not be confused with
the further question : Does this evidence warrant us
in assuming the truth of causal laws in the future
and in unobserved portions of the past? For the
present, I am only asking what are the grounds which

220 SCIENTIFIC METHOD IN PHILOSOPHY

lead to a belief in causal laws, not whether these
grounds axe adequate to support the belief in universal
causation.

The first step is the discovery of approximate un-
analysed uniformities of sequence or coexistence.
After lightning comes thunder, after a blow received
comes pain, after approaching a fire comes warmth ;
again, there are uniformities of coexistence, for ex-
ample between touch and sight, between certain
sensations in the throat and the sound of one's own
voice, and so on. Every such uniformity of sequence
or coexistence, after it has been experienced a certain
number of times, is followed by an expectation that it
will be repeated on future occasions, i.e. that where one
of the correlated events is found, the other will be
found also. The connection of experienced past
uniformity with expectation as to the future is just
one of those uniformities of sequence which we have
observed to be true hitherto. This affords a psycho-
logical account of what may be called the animal belief
in causation, because it is somet-hing which can be
observed in horses and dogs, and is rather a habit of
acting than a real belief. So far, we have merely
repeated Hume, who carried the discussion of cause up
to this point, but did not, apparently, perceive how
much remained to be said.

Is there, in fact, any characteristic, such as might
be called causality or uniformity, which is found to
hold throughout the observed past ? And if so, how
is it to be stated ?

The particular uniformities which we mentioned
before, such as lightning being followed by thunder,
are not found to be free from exceptions. We some-
times see lightning without hearing thunder ; and
although, in such a case, we suppose that thunder

ON THE NOTION OF CAUSE 221

might have been heard if we had been nearer to the
lightning, that is a supposition based on theory, and
therefore incapable of being invoked to support the
theory. What does seem, however, to be shown by
scientific experience is this : that where an observed
uniformity fails, some wider uniformity can be found,
embracing more circumstances, and subsuming both
the successes and the failures of the previous uniform-
ity. Unsupported bodies in air fall, unless they are
balloons or aeroplanes ; but the principles of mechanics
give uniformities which apply to balloons and aero-
planes just as accurately as to bodies that fall. There
is much that is hypothetical and more or less artificial
in the uniformities affirmed by mechanics, because,
when they cannot otherwise be made applicable,
unobserved bodies are inferred in order to account for
observed peculiarities. Still, it is an empirical fact
that it is possible to preserve the laws by assuming
such bodies, and that they never have to be assumed
in drcumstances in which they ought to be observable.
Thus the empirical verification of mechanical laws
may be admitted, although we must also admit that
it is less complete and triumphant than is sometimes
supposed.

Assuming now, what must be admitted to be doubt-
ful, that the whole of the past has proceeded according
to invariable laws, what can we say as to the nature
of these laws ? They will not be of the simple type
which asserts that the same cause always produces
the same effect. We may take the law of gravitation
as a sample of the kind of law that appears to be
verified without exception. In order to state this
law in a form which observation can confirm, we will
confine it to the solar system. It then states that the
motions of planets and their satellites have at every

222 SCIENTIFIC METHOD IN PHILOSOPHY

instant an acceleration compounded of accelerations
towards all the other bodies in the solar system,
proportional to the matters of these bodies and
inversely proportional to the squares of their distances.
In virtue of this law, given the state of the solar system
throughout any finite time, however short, its state at
all earlier and later times is determinate except in so
far as other forces than gravitation or other bodies
than those in the solar system have to be taken into
consideration. But other forces, so far as science
can discover, appear to be equally regular, and equally
capable of being summed up in single causal laws. If
the mechanical account of matter were complete, the
whole physical history of the universe, past and
future, could be inferred from a sufficient number of
data concerning an assigned finite time, however
short.

In the mental world, the evidence for the universality
of causal laws is less complete than in the physical
world. Psychology cannot boast of any triumph
comparable to gravitational astronomy. Nevertheless-
the evidence is not very greatly less than in the physical
world. The crude and approximate causal laws from
which science starts are just as easy to discover in the
mental sphere as in the physical. In the world of
sense, there axe to begin with the correlations of sight
and touch and so on, and the facts which lead us to
connect various kinds of sensations with eyes, ears,
nose, tongue, etc. Then there are such facts as that
our body moves in answer to our volitions. Excep-
tions exist, but are capable of being explained as
easily as the exceptions to the rule that unsupported
bodies in air fall. There is, in fact, just such a degree
of evidence for causal laws in psychology as will
warrant the psychologist in assuming them as a matter

ON THE NOTION OF CAUSE 223

of course, though not such a degree as will suffice to
remove all doubt from the mind of a sceptical inquirer.
It should be observed that causal laws in which the
given term is mental and the inferred term physical,
or vice versa, are at least as easy to discover as causal
laws in which both terms are mental.

It will be noticed that, although we have spoken of

causal laws, we have not hitherto introduced the

word " cause." At this stage, it will be well to say a

few words on legitimate and illegitimate uses of this

word. The word " cause," in the scientific account of

the world, belongs only to the early stages, in which

small preliminary, approximate generalizations are

being ascertained with a view to subsequent larger

and more invariable laws. We may say "Arsenic

causes death," so long as we are ignorant of the precise

process by which the result is brought about. But in

a sufficiently advanced science, the word "cause"

will not occur in any statement of invariable laws.

There is, however, a somewhat rough and loose use

of the word " cause " which may be preserved. The

approximate uniformities which lead to its pre-scientific

employment may turn out to be true in all but very

rare and exceptional circumstances, perhaps in all

circumstances that actually occur. In such cases, it

is convenient to be able to speak of the antecedent

event as the " cause " and the subsequent event as

the " effect." In this sense, provided it is realized

that the sequence is not necessary and may have

exceptions, it is still possible to employ the words

" cause " and " effect." It is in this sense, and in this

sense only, that we shall intend the words when we

speak of one particular event "causing" another

particular event, as we must sometimes do if we are

to avoid intolerable drcumlocution.

224 SCIENTIFIC METHOD IN PHILOSOPHY

III. We come now to our third question, namely :
What reason can be given for believing that causal
laws will hold in future, or that they have held in
unobserved portions of the past ?

What we have said so far is that there have been
hitherto certain observed causal laws, and that all the
empirical evidence we possess is compatible with the
view that everything, both mental and physical, so
far as our observation has extended, has happened in
accordance with causal laws. The law of universal
causation, suggested by these facts, may be enunciated
as follows :

" There are such invariable relations between differ-
ent events at the same or different times that, given
the state of the whole universe throughout any finite
time, however short, every previous and subsequent
event can theoretically be determined as a function
of the given events during that time."

Have we any reason to believe this universal law ?
Or, to ask a more modest question, have we any
reason to believe that a particular causal law, such as
the law of gravitation, will continue to hold in the
future ?

Among observed causal laws is this, that observation
of uniformities is followed by expectation of their
recurrence. A horse who has been driven always
along a certain road expects to be driven along that
road again ; a dog who is always fed at a certain hour
expects food at that hour and not at any other. Such
expectations, as Hume pointed out, explain only too
well the common-sense belief in uniformities of se-
quence, but they afford absolutely no logical ground
for beliefs as to the future, not even for the belief
that we shall continue to expect the continuation of
experienced uniformities, for that is precisely one of

ON THE NOTION OF CAUSE 225

those causal laws for which a ground has to be sought.
If Hume's account of causation is the last word,
we have not only no reason to suppose that the
sun will rise to-morrow, but no reason to suppose
that five minutes hence we shall still expect it to rise
to-morrow.

It may, of course, be said that all inferences as to
the future are in fact invalid, and I do not see how
such a view could be disproved. But, while admitting
the legitimacy of such a view, we may nevertheless
inquire : If inferences as to the future are valid, what
principle must be involved in making them ?

The principle involved is the principle of induction, 1
which, if it is true, must be an a priori logical law, not
capable of being proved or disproved by experience.
It is a difficult question how this principle ought to be
formulated ; but if it is to warrant the inferences
which we wish to make by its means, it must lead to
the following proposition : " If, in a great number of
instances, a tiling of a certain kind is associated in a
certain way with a thing of a certain other kind, it is
probable that a thing of the one kind is always similarly
associated with a thing of the other kind ; and as the
number of instances increases, the probability ap-
proaches indefinitely near to certainty." It may
well be questioned whether this proposition is true ;
but if we admit it, we can infer that any characteristic
of the whole of the observed past is likely to apply
to the future and to the unobserved past. This
proposition, therefore, if it is true, will warrant the
inference that causal laws probably hold at all times,
future as well as past; but without this principle,
the observed cases of the truth of causal laws afford

xOn this subject, see Keynes's Treatise on Probability
t 1921).

15

226 SCIENTIFIC METHOD IN PHILOSOPHY

no presumption as to the unobserved cases, and
therefore the existence of a thing not directly observed
can never be validly inferred.

It is thus the principle of induction, rather than the
law of causality, which is at the bottom of all inferences
as to the existence of things not immediately given.
With the principle of induction, all that is wanted for
such inferences can be proved; without it, all such
inferences are invalid. This principle has not received
the attention which its great importance deserves.
Those who were interested in deductive logic naturally
enough ignored it, while those who emphasized the
scope of induction wished to maintain that all logic
is empirical, and therefore could not be expected to
realize that induction itself, their own darling, required
a logical principle which obviously could not be proved
inductively, and must therefore be a priori if it could
be known at all.

The view that the law of causality itself is a priori
cannot, I think, be maintained by anyone who rftfl.1i7.e3
what a complicated principle it is. In the form
which states that " every event has a cause " it looks
simple ; but on examination, " cause " is merged in
" causal law," and the definition of a " causal law " is
found to be far from simple. There must necessarily
be some a priori principle involved in inference from
the existence of one thing to that of another, if such
inference is ever valid ; but it would appear from the
above analysis that the principle in question is induc-
tion, not causality. Whether inferences from past to
future are valid depends wholly if our discussion has
been sound, upon the inductive principle : if it is true,
such inferences are valid, and if it is false, they are
invalid.

IV. I come now to the question how the conception

ON THE NOTION OF CAUSE 227

of causal laws which we have arrived at is related to
the traditional conception of cause as it occurs in
philosophy and common sense.

Historically, the notion of cause has been bound
up with that of human volition. The typical cause
would be the fiat of a king. The cause is supposed
to be " active," the effect " passive." From this it
is easy to pass on to the suggestion that a " true "
cause must contain some prevision of the effect ;
hence the effect becomes the " end " at which the
cause aims, and teleology replaces causation in the
explanation of nature. But all such ideas, as applied
to physics, are mere anthropomorphic superstitions.
It is as a reaction against these errors that Mach and
others have urged a purely " descriptive " view of
physics : physics, they say, does not aim at telling us
" why " things happen, but only " how " they happen.
And if the question " why ? " means anything more
than the search for a general law according to which
a phenomenon occurs, then it is certainly the case
that this question cannot be answered in physics and
ought not to be asked. In this sense, the descriptive
view is indubitably in the right. But in using causal
laws to support inferences from the observed to the
unobserved, physics ceases to be pwdy descriptive,
and it is these laws which give the scientifically useful
part of the traditional notion of " cause." There is
therefore something to preserve in this notion, though
it is a very tiny part of what is commonly assumed in
orthodox metaphysics.

In order to understand the difference between the
kind of cause which science uses and the kind which
we naturally imagine, it is necessary to shut out, by
an effort, everything that differentiates between past
and future. This is an extraordinarily difficult thing

228 SCIENTIFIC METHOD IN PHILOSOPHY

to do, because our mental life is so intimately bound
up with difference Not only do memory and hope
make a difference in our feelings as regards past and
future, but almost our whole vocabulary is filled with
the idea of activity, of things done now for the sake
of their future effects. All transitive verbs involve
the notion of cause as activity, and would have to be
replaced by some cumbrous periphrasis before this
notion could be eliminated.

Consider such a statement as, " Brutus killed Caesar."
On another occasion, Brutus and Caesar might engage
our attention, but for the present it is the killing that
we have to study. We may say that to kill a person
is to cause his death intentionally. This means that
desire for a person's death causes a certain act, because
it is believed that that act will cause the person's
death ; or more accurately, the desire and the belief
jointly cause the act. Brutus desires that Caesar
should be dead, and believes that he will be dead if
he is stabbed ; Brutus therefore stabs him, and the
stab causes Caesar's death, as Brutus expected it would.
Every act which realizes a purpose involves two causal
steps in this way : C is desired, and it is believed
(truly if the purpose is achieved) that B will cause C ;
the desire and the belief together cause B, which in
turn causes C. Thus we have first A, which is a desire
for C and a belief that B (an act) will cause C ; then
we have B, the act caused by A, and believed to be
a cause of C ; then, if the belief was correct, we have
C, caused by B, and if the belief was incorrect we have
disappointment. Regarded purely scientifically, this
series A, B, C may equally well be considered hi the
inverse order, as they would be at a coroner's inquest.
But from the point of view of Brutus, the desire, which
comes at the beginning, is what makes the whole

ON THE NOTION OF CAUSE 229

series interesting. We feel that if his desires had been
different, the effects which he in fact produced would
not have occurred. This is true, and gives him a
sense of power and freedom. It is equally true that
if the effects had not occurred, his desires would have
been different, since being what they were the effects
did occur. Thus the desires are determined by their
consequences just as much as the consequences by the
desires ; but as we cannot (in general) know in advance
the consequences of our desires without knowing our
desires, this form of inference is uninteresting as applied
to our own acts, though quite vital as applied to those
of others.

A cause, considered scientifically, has none of that
analogy with volition which makes us imagine that the
effect is compelled by it. A cause is an event or group
of events of some known general character, and having
a known relation to some other event, called the
effect ; the relation being of such a kind that only one
event, or at any rate only one well-defined sort of
event, can have the relation to a given cause. It is
customary only to give the name "effect" to an
event which is later than the cause, but there is no
kind of reason for this restriction. We shall do better
to allow the effect to be before the cause or simultane-
ous with it, because nothing of any scientific importance
depends upon its being after the cause.

If the inference from cause to effect is to be indubit-
able, it seems that the cause can hardly stop short of
the whole universe. So long as anything is left out,
something may be left out which alters the expected
result. But for practical and scientific purposes,
phenomena can be collected into groups which are
causally self-contained, or nearly so. In the common
notion of causation, the cause is a single event we say

230 SCIENTIFIC METHOD IN PHILOSOPHY

the lightning causes the thunder, and so on. But it is
difficult to know what we mean by a single event ; and
it generally appears that, in order to have anything
approaching certainty concerning the effect, it is
necessary to include many more circumstances in the
cause than unscientific common sense would suppose.
But often a probable causal connection, where the
cause is fairly simple, is of more practical importance
than a more indubitable connection in which the
cause is so complex as to be hard to ascertain.

To sum up: the strict, certain, universal law of
causation which philosophers advocate is an ideal,
possibly true, but not known to be true in virtue of
any available evidence. What is actually known, as
a matter of empirical science, is that certain constant
relations are observed to hold between the members
of a group of events at certain times, and that when
such relations fail, as they sometimes do, it is usually
possible to discover a new, more constant relation
by enlarging the group. Any such constant relation
between events of specified kinds with given intervals
of time between them is a "causal law." But all
causal laws are liable to exceptions, if the cause is
less than the whole state of the universe ; we believe,
on the basis of a good deal of experience, that such
exceptions can be dealt with by enlarging the group
we caJl the cause, but this belief, wherever it is still
unverified, ought not to be regarded as certain, but
only as suggesting a direction for further inquiry.

A very common causal group consists of volitions
and the consequent bodily acts, though exceptions
arise (for example) through sudden paralysis. Another
very frequent connection (though here the exceptions
are much more numerous) is between a bodily act and
the realization of the purpose which led to the act.

ON THE NOTION OF CAUSE 231

These connections are patent, whereas the causes of
desires are more obscure. Thus it is natural to begin
causal series with desires, to suppose that all causes
are analogous to desires, and that desires themselves
arise spontaneously. Such a view, however, is not
one which any serious psychologist would maintain.
But this brings us to the question of the application
of our analysis of cause to the problem of free will

V. The problem of free will is so intimately bound
up with the analysis of causation that, old as it is, we
need not despair of obtaining new light on it by the
help of new views on the notion of cause. The free-will
problem has, at one time or another, stirred men's
passions profoundly, and the fear that the will might
not be free has been to some men a source of great
unhappiness. I believe that, under the influence of a
cool analysis, the doubtful questions involved will be
found to have no such emotional importance as is
sometimes thought, since the disagreeable conse-
quences supposed to flow from a denial of free will do
not flow from this denial in any form in which there
is reason to make it. It is not, however, on this
account chiefly that I wish to discuss this problem,
but rather because it affords a good example of the
clarifying effect of analysis and of the interminable
controversies which may result from its neglect.

Let us first try to discover what it is we really
desire when we desire free will Some of our reasons
for desiring free will are profound, some trivial. To
begin with the former : we do not wish to fed ourselves
in the hands of fate, so that, however much we may
desire to will one thing, we may nevertheless be com-
pelled by an outside force to will another. We do
not wish to think that, however much we may desire
ty> act well, heredity and surroundings may f ores us

232 SCIENTIFIC METHOD IN PHILOSOPHY

into acting ill. We wish to feel that, in cases of
doubt, our choice is momentous and lies within our
power. Besides these desires, which are worthy of
all respect, we have, however, others not so respectable,
which equally make us desire free will. We do not
like to think that other people, if they knew enough,
could predict our actions, though we know that we
can cf ten predict those of other people, especially if
they are elderly. Much as we esteem the old gentle-
man who is our neighbour in the country, we know
that when grouse are mentioned he will tell the story
of the grouse in the gun-room. But we ourselves are
not so mechanical : we never tell an anecdote to the
same person twice, or even once unless he is sure to
enjoy it ; although we once met (say) Bismarck, we
are quite capable of hearing him mentioned without
relating the occasion when we met him. In this
sense, everybody thinks that he himself has free will,
though he knows that no one else has. The desire
for .this kind of free will seems to be no better than a
form of vanity. I do not believe that this desire can
be gratified with any certainty ; but the other, more
respectable desires are, I believe, not inconsistent
with any tenable form of determinism.

We have thus two questions to consider : (i) Are
human actions theoretically predictable from a suffi-
cient number of antecedents ? (2) Are human actions
subject to an external compulsion ? The two ques-
tions, as I shall try to show, are entirely distinct, and
we may answer the first in the affirmative without
therefore being forced to give an affirmative answer
to the second.

(i) Are human actions theoretically predictable from
a sufficient number of antecedents? Let us first en-
deavour to give precision to this question. We may

ON THE NOTION OF CAUSE 233

state the question thus : Is there some constant relation
between an act and a certain number of earlier events,
such that, when the earlier events are given, only one
act, or at most only acts with some well-marked
character, can have this relation to the earlier events ?
If this is the case, then, as soon as the earlier events
are known, it is theoretically possible to predict either
the precise act, or at least the character necessary to
its fulfilling the constant relation.

To this question, a negative answer has been given
by Bergson, in a form which calls in question the
general applicability of the law of causation. He
maintains that every event, and more particularly
every mental event, embodies so much of the past
that it could not possibly have occurred at any earlier
time, and is therefore necessarily quite different from
all previous and subsequent events. If, for example,
I read a certain poem many times, my experience on
each occasion is modified by the previous readings,
and my emotions are never repeated exactly. The
principle of causation, according to him, asserts that
the same cause, if repeated, will produce the same
effect. But owing to memory, he contends, this
principle does not apply to mental events. What is
apparently the same cause, if repeated, is modified
by the mere fact of repetition, and cannot produce
the same effect. He infers that every mental event
is a genuine novelty, not predictable from the past,
because the past contains nothing exactly like it by
which we could imagine it. And on this ground he
regards the freedom of the will as unassailable.

Bergson's contention has undoubtedly a great deal
of truth, and I have no wish to deny its importance.
But I do not think its consequences are quite what
he believes them to be. It is not necessary for the

234 SCIENTIFIC METHOD IN PHILOSOPHY

detemunist to maintain that he can foresee the whole
particularity of the act which will be performed. If
he could foresee that A was going to murder B, his
foresight would not be invalidated by the fact that he
could not know all the infinite complexity of A's state
of mind in committing the murder, nor whether the
murder was to be performed with a knife or with a
revolver. .If the kind of act which will be performed
can be foreseen within narrow limits, it is of little
practical interest that there are fine shades which
cannot be foreseen. No doubt every time the story
of the grouse in the gun-room is told, there will be
slight differences due to increasing habitualness, but
they do not invalidate the prediction that the story
will be told. And there is nothing in Bergson's
argument to show that we can never predict what
kind of act will be performed.

Again, his statement of the law of causation is
inadequate. The law does not state merely that, if
the same cause is repeated, the same effect will result.
It states rather that there is a constant relation between
causes of certain kinds and effects of certain kinds.
For example, if a body falls freely, there is a constant
relation between the height through which it falls and
the time it takes in falling. It is not necessary to have
a body fall through the same height which has been
previously observed, in order to be able to foretell the
length of time occupied in falling. If this were
necessary, no prediction would be possible, since it
would be impossible to make the height exactly the
same on two occasions. Similarly, the attraction
which the sun will exert on the earth is not only known
at distances for which it has been observed, but at all
distances, because it is known to vary as the inverse
square of the distance, III f act, what i foun4 to b?

ON THE NOTION OF CAUSE 235

repeated is always the relation of cause and effect, not
the cause itself ; all that is necessary as regards the
cause is that it should be of the same kind (in the
relevant respect) as earlier causes whose effects have
been observed.

Another respect in which Bergson's statement of
causation is inadequate is in its assumption that the
cause must be one event, whereas it may be two or
more events, or even some continuous process. The
substantive question at issue is whether mental events
are determined by the past. Now in such a case as
the repeated reading of a poem, it is obvious that our
feelings in reading the poem are most emphatically
dependent upon the past, but not upon one single
event in the past. All our previous readings of the
poem must be included in the cause. But we easily
perceive a certain law according to which the effect
varies as the previous readings increase in number,
and in fact Bergson himself tacitly assumes such a
law. We decide at last not to read the poem again,
because we know that this time the effect would be
boredom. We may not know all the niceties and
shades of the boredom we should fed, but we know
enough to guide our decision, and the prophecy of
boredom is none the less true for being more or less
general Thus the kinds of cases upon which Bergson
relies are insufficient to show the impossibility of
prediction in the only sense in which prediction has
practical or emotional interest. We may therefore
leave the consideration of his arguments and address
ourselves to the problem directly.

The law of causation, according to which later
events can theoretically be predicted by means of
earlier events, has often been held to be a priori, a
necessity of thought, a category without which science

236 SCIENTIFIC METHOD IN PHILOSOPHY

would be impossible. These claims seem to me
excessive. In certain directions the law has been
verified empirically, and in other directions there is
no positive evidence against it. But science can use it
where it has been found to be true, without being
forced into any assumption as to its truth in other
fields. We cannot, therefore, feel any a priori cer-
tainty that causation must apply to human volitions.

The question how far human volitions are subject to
causal laws is a purely empirical one. Empirically it
seems plain that the great majority of our volitions
have causes, but it cannot, on this account, be hdd
necessarily certain that all have causes. There axe,
however, precisely the same kinds of reasons for
regarding it as probable that they all have causes as
there are in the case of physical events.

We may suppose though this is doubtful that
there are laws of correlation of the mental and the
physical, in virtue of which, given the state of all the
matter in the world, and therefore of all the brains and
living organisms, the state of all the minds in the
world could be inferred, while conversely the state of
all the matter in the world could be inferred if the
state of all the minds were given. It is obvious that
there is some degree of correlation between brain and
mind, and it is impossible to say how complete it
may be. This, however, is not the point which I
wish to elicit. What I wish to urge is that, even if
we admit the most extreme claims of determinism
and of correlation of mind and brain, still the conse-
quences inimical to what is worth preserving in free
will do not follow. The belief that they follow results,
I think, entirely from the assimilation of causes to
volitions, and from the notion that causes compel
their effects in some sense analogous to that in which

ON THE NOTION OF CAUSE 237

a human authority can compel a man to do what he
would rather not do. This assimilation, as soon as
the true nature of scientific causal laws is realized, is
seen to be a sheer mistake. But this brings us to the
second of the two questions which we raised in regard
to free will, namely whether, assuming determinism,
our actions can be in any proper sense regarded as
compelled by outside forces.

(2) Are human actions subject to an external com-
pulsion t We have, in deliberation, a subjective sense
of freedom, which is sometimes alleged against the
view that volitions have causes. This sense of freedom,
however, is only a sense that we can choose which
we please of a number of alternatives : it does not
show us that there is no causal connection between
what we please to chose and our previous history.
The supposed inconsistency of these two springs from
the habit of conceiving causes as analogous to volitions
a habit which often survives unconsciously in those
who intend to conceive causes in a more scientific
manner. If a cause is analogous to a volition, outside
causes will be analogous to an alien will, and acts
predictable from outside causes will be subject to
compulsion. But this view of cause is one to which
science lends no countenance. Causes, we have seen,
do not compel their effects, any more than effects
compel their causes. There is a mutual relation, so
that either can be inferred from the other. When
the geologist infers the past state of the earth from
its present state, we should not say that the present
state compels the past state to have been what it
was ; yet it renders it necessary as a consequence of
the data, in the only sense in which effects are rendered
necessary by their causes. The difference which we
feel, in this respect, between causes and effects is a

238 SCIENTIFIC METHOD IN PHILOSOPHY

mere confusion due to the fact that we remember past
events but do not happen to have memory of the
future.

The apparent indeterminateness of the future, upon
which some advocates of free will rely, is merely a
result of our ignorance. It is plain that no desirable
kind of free will can be dependent simply upon our
ignorance ; for if that were the case, animals would
be more free than men, and savages than civilized
people. Free will in any valuable sense must be
compatible with the fullest knowledge. Now, quite
apart from any assumption as to causality, it is obvious
that complete knowledge would embrace the future
as well as the past. Our knowledge of the past is not
wholly based upon causal inferences, but is partly
derived from memory. It is a mere accident that we
have no memory of the future. We might as in the
pretended visions of seers see future events immedi-
ately, in the way in which we see past events. They
certainly will be what they will be, and are in this
sense just as determined as the past. If we saw
future events in the same immediate way in which
we see past events, what kind of free will would still
be possible ? Such a kind would be wholly indepen-
dent of determinism : it could not be contrary to even
the most entirely universal reign of causality. And
such a kind must contain whatever is worth having in
free will, since it is impossible to believe that mere
ignorance can be the essential condition of any good
thing. Let us therefore imagine a set of beings who
know the whole future with absolute certainty, and
let us ask ourselves whether they could have anything
that we should call free will.

Such beings as we are imagining would not have to
wait for the event in order to know what decision

ON THE NOTION OF CAUSE 239

they were going to adopt on some future occasion.
They would know now what their volitions were
going to be. But would they have any reason to
regret this knowledge ? Surely not, unless the fore-
seen volitions were in themselves regrettable. And it
is less likely that the foreseen volitions would be
regrettable if the steps which would lead to them
were also foreseen. It is difficult not to suppose that
what is foreseen is fated, and must happen however
much it may be dreaded. But human actions are the
outcome of desire, and no foreseeing can be true
unless it takes account of desire. A foreseen volition
will have to be one which does not become odious
through being foreseen. The beings we are imagining
would easily come to know the causal connections of
volitions, and therefore their volitions would be better
calculated to satisfy their desires than ours are. Since
volitions are the outcome of desires, a prevision of
volitions contrary to desires could not- be a true one.
It must be remembered that the supposed prevision
would not create the future any more thaa memory
creates the past. We do not think we were necessarily
not free in the past, merely because we can now
remember our past volitions. Similarly, we might be
free in the future, even if we could now see what our
future volitions were going to be. Freedom, in short,
in any valuable sense, demands only that our volitions
shall be, as they are, the result of our own desires, not
of an outside force compelling us to will what we would
rather not will. Everything else is confusion of
thought, due to the feeling that knowledge compels
the happening of what it knows when this is future,
though it is at once obvious that knowledge has no
such power in regard to the past. Free will, therefore,
is true in the only form which is important ; and the

240 SCIENTIFIC METHOD IN PHILOSOPHY

desire for other forms is a mere effect of insufficient
analysis.

What has been said on philosophical method in the
foregoing lectures has been rather by means of illus-
trations in particular cases than by means of general
precepts. Nothing of any value can be said on
method except through examples; but now, at the
end of our course, we may collect certain general
maxims which may possibly be a help in acquiring a
philosophical habit of mind and a guide in looking for
solutions of philosophic problems.

Philosophy does not become scientific by making
use of other sciences, in the kind of way in which, e.g.
Herbert Spencer does. Philosophy aims at what is
general, and the special sciences, however they may
suggest large generalizations, cannot make them certain.
And a hasty generalization, such as Spencer's general-
ization of evolution, is none the less hasty because
what is generalized is the latest scientific theory.
Philosophy is a study apart from the other sciences :
its results cannot be established by the other sciences,
and conversely must not be such as some other science
might conceivably contradict. Prophecies as to the
future of the universe, for example, are not the business
of philosophy; whether the universe is progressive,
retrograde, or stationary, it is not for the philosopher
to say.

In order to become a scientific philosopher, a certain
peculiar mental discipline is required. There must be
present, first of aU, the desire to know philosophical
truth, and this desire must be sufficiently strong to
survive through years when there seems no hope of
its finding any satisfaction. The desire to know
philosophical truth is very rare in its purity, it is

ON THE NOTION OF CAUSE 241

not often found even among philosophers. It is
obscured sometimes particularly after long periods of
fruitless search by the desire to think we know.
Some plausible opinion presents itself, and by turning
our attention away from the objections to it, or merely
by not making great efforts to find objections to it,
we may obtain the comfort of believing it, although,
if we had resisted the wish for comfort, we should
have come to see that the opinion was false. Again
the desire for unadulterated truth is often obscured,
in professional philosophers, by love of system : the
one little fact which will not come inside the philoso-
pher's edifice has to be pushed and tortured until it
seems to consent. Yet the one little fact is more
likely to be important for the future than the system
with which it is inconsistent. Pythagoras invented a
system which fitted admirably with all the facts he
knew, except the incommensurability of the diagonal
of a square and the side ; this one little fact stood out,
and remained a fact even after Hippasos of Metapon-
tion was drowned for revealing it. To us, the discovery
of this fact is the chief claim of Pythagoras to immor-
tality, while his system has become a matter of merely
historical curiosity. 1 Love of system, therefore, and
the system-maker's vanity which becomes associated
with it, are among the snares that the student of
philosophy must guard against.

The desire to establish this or that result, or generally
to discover evidence for agreeable results, of whatever
kind, has of course been the chief obstacle to honest
philosophizing. So strangely perverted do men become
by unrecognized passions, that a determination in

'The above remarks, for purposes of illustration, adopt
one of several possible opinions on each of several disputed
points.

16

SCIENTIFIC METHOD IN PHILOSOPHY

advance to arrive at this or that conclusion is generally
regarded as a mark of virtue, and those whose studies
lead to an opposite conclusion are thought to be wicked.
No doubt it is commoner to wish to arrive at an agree-
able result than to wish to arrive at a true result.
But only those in whom the desire to arrive at a true
result is paramount can hope to serve any good purpose
by the study of philosophy.

But even when the desire to know exists in the requisite
strength, the mental vision by which abstract truth
is recognized is hard to distinguish from vivid imagin-
ability and consonance with mental habits. It is
necessary to practise methodological doubt, like
Descartes, in order to loosen the hold of mental habits ;
and it is necessary to cultivate logical imagination, in
order to have a number of hypotheses at command,
and not to be the slave of the one which common
sense has rendered easy to imagine. These two
processes, of doubting the familiar and imagining the
unfamiliar, are correlative, and form the chief part of
the mental training required for a philosopher.

The naive beliefs which we find in ourselves when
we first begin the process of philosophic reflection
may turn out, in the end, to be almost all capable of
a true interpretation; but they ought all, before
being admitted into philosophy, to undergo the ordeal
of sceptical criticism. Until they have gone through
this ordeal, they are mere blind habits, ways of be-
having rather than intellectual convictions. And
although it may be that a majority will pass the
test, we may be pretty sure that some will not, and
that a serious readjustment of our outlook ought to
result. In order to break the dominion of habit, we
must do our best to doubt the senses, reason, morals,
everything in short. In some directions, doubt will

ON THE NOTION OF CAUSE 243

be found possible ; in others, it will be checked by
that direct vision of abstract truth upon which the
possibility of philosophical knowledge depends.

At the same time, and as an essential aid to the
direct perception of the truth, it is necessary to acquire
fertility in imagining abstract hypotheses. This is,
I think, what has most of all been lacking hitherto in
philosophy. So meagre was the logical apparatus
that all the hypotheses philosophers could imagine
were found to be inconsistent with the facts. Too
often this state of things led to the adoption of heroic
measures, such as a wholesale denial of the facts,
when an imagination better stocked with logical tools
would have found a key to unlock the mystery. It is
in this way that the study of logic becomes the central
study in philosophy : it gives the method of research
in philosophy, just as mathematics gives the method
in physics. And as physics, which, from Plato to the
Renaissance, was as unprogressive, dim, and supersti-
tious as philosophy, became a science through Galileo's
fresh observation of facts and subsequent mathematical
manipulation, so philosophy, in our own day, is
becoming scientific through the simultaneous acquisi-
tion of new facts and logical methods.

In spite, however, of the new possibility of progress
in philosophy, the first effect, as in the case of physics,
is to diminish very greatly the extent of what is
thought to be known. Before Galileo, people believed
themselves possessed of immense knowledge on all
the most interesting questions in physics. He estab-
lished certain facts as to the way in which bodies fall,
not very interesting on their own account, but of
quite immeasurable interest as examples of real
knowledge and of a new method whose future fruitful-
ness he himself divined. But his few facts sufficed to

244 SCIENTIFIC METHOD IN PHILOSOPHY

destroy the whole vast system of supposed knowledge
handed down from Aristotle, as even the palest
morning sun suffices to extinguish the stars. So in
philosophy: though some have believed one system,
and others another, almost all have been of opinion
that a great deal was known ; but all this supposed
knowledge in the traditional systems must be swept
away, and a new beginning must be made, which we
shall esteem fortunate indeed if it can attain results
comparable to Galileo's law of falling bodies.

By the practice of methodological doubt, if it is
genuine and prolonged, a certain humility as to our
knowledge is induced: we become glad to know
anything in philosophy, however seemingly trivial.
Philosophy has suffered from the lack of this kind of
modesty. It has made the mistake of attacking the
interesting problems at once, instead of proceeding
patiently and slowly, accumulating whatever solid
knowledge was obtainable, and trusting the great
problems to the future. Men of science are not
ashamed of what is intrinsically trivial, if its conse-
quences are likely to be important; the immediate
outcome of an experiment is hardly ever interesting
on its own account. So in philosophy, it is often
desirable to expend time and care on matters which,
judged alone, might seem f rivolous, for it is often only
through the consideration of such matters that the
greater problems can be approached.

When our problem has been selected, and the
necessary mental discipline has been acquired, the
method to be pursued is fairly uniform. The big
problems which provoke philosophical inquiry are
found, on examination, to be complex, and to depend
upon a number of component problems, usually more
abstract than those of which they are the components.

ON THE NOTION OF CAUSE 245

It will generally be found that all our initial data, all
the facts that we seem to know to begin with, suffer
from vagueness, confusion, and complexity. Current
philosophical ideas share these defects ; it is therefore
necessary to create an apparatus of precise conceptions
as general and as free from complexity as possible,
before the data can be analysed into the kind of
premisses which philosophy aims at discovering. In
this process of analysis, the source of difficulty is
tracked further and further back, growing at each
stage more abstract, more refined, more difficult to
apprehend. Usually it will be found that a number
of these Artra.nHiTifl.ri1y abstract questions underlie
any one of the big obvious problems. When every-
thing has been done that can be done by method, a
stage is reached where only direct philosophic vision
can carry matters further. Here only genius will
avail. What is wanted, as a rule, is some new effort
of logical imagination, some glimpse of a possibility
never conceived before, and then the direct perception
that this possibility is realized in the case in question.
Failure to thinfe of the right possibility leaves insoluble
difficulties, balanced arguments pro and con, utter
bewilderment and despair. But the right possibility,
as a rule, when once conceived, justifies itself swiftly
by its astonishing power of absorbing apparently
conflicting facts. From this point onward, the work
of the philosopher is synthetic and comparatively easy ;
it is in the very last stage of the analysis that the real
difficulty consists.

Of the prospect of progress in philosophy, it would
be rash to speak with confidence. Many of the
traditional problems of philosophy, perhaps most of
those which have interested a wider circle than that
of technical students, do not appear to be soluble by

246 SCIENTIFIC METHOD IN PHILOSOPHY

scientific methods. Just as astronomy lost much of
its human interest when it ceased to be astrology, so
philosophy must lose in attractiveness as it grows
less prodigal of promises. But to the large and still
growing body of men engaged in the pursuit of science
men who hitherto, not without justification, have
turned aside from philosophy with a certain contempt
the new method, successful already in such time-
honoured problems as number, infinity, continuity,
space and time, should make an appeal which the
older methods have wholly failed to make. Physics,
with its principle of relativity and its revolutionary
investigations into the nature of matter, is feeling the
need for that kind of novelty in fundamental hypo-
theses which scientific philosophy aims at facilitating.
The one and only condition, I believe, which is neces-
sary in order to secure for philosophy in the near future
an achievement surpassing all that has hitherto been
accomplished by philosophers, is the creation of a
school of men with scientific training and philosophical
interests, unhampered by the traditions of the past,
and not misled by the literary methods of those who
copy the ancients in all except their merits.

INDEX

Absolute, 16, 48.
Abstraction, principle of, 51,

132 ff.
Achilles, Zeno's argument of,

178.

Acquaintance, 35, 151.
Activity, 228 ff.
Allman, 16571.
Analysis, 189, 209, 214, 245.
legitimacy of, 156.

An gnrimflnder, 13.

Antinomies, Kant's, 159 ff.

Aquinas, 20.

Aristotle, 49, 164 n., 1652.,

244.
Arrow, Zeno's argument of,

178.

Assertion, 61.
Atomism, logical, 14-
Atomists, 164.

Belief, 67.
primitive and derivative,

75 ff.
Bergson, 14, 21, 23, 29 ff., 143,

144, 157, 162, 169, 179,

184, 233 ff.

Berkeley, 63, 64, 102.
Bolzano, 169.
Boole, 50.

Bradley, 16, 48, 170.
Broad, 131. 17?
Brochard, 174 **
Burnet, 29 n., 164*., i66.,

175 i->

Calderon, 103.

Cantor, Georg, 8, 9, 159, 169,

I94 199. 204.
Cantor, Moritz, 173 n.
Categories, 48.
Causal laws, 115, 215 ff.

evidence for, 219 ff.

in psychology, 222.
Causation, 43 ff., 86, 215 ff.

law of, 224.

not a priori, 226, 235.
Cause, 223, 226.
Certainty, degrees of, 74, 75,

219.
Change, demands analysis,

158-

Cinematograph, 154, 179*
Classes, 206.

non-existence of, 2ioff.
Classical tradition, 14 ff., 68.
Complexity, 152, 162 ff.
Compulsion, 232, 2365.
Congruence, 199.
Consecutiveness, 140.
Conservation, no.
Constituents of facts, 60, 150.
Construction v. inference, 8.
Contemporaries, initial, 125.
Continuity, 70, 135 ff., 147 ff"

159 ff-

of change, in, 113. 136 ff.
Correlation of mental and

physical, 235.
Counting, 169, 187. 192 ff-

207.
Couturat, 49

Dante, 20.

248 SCIENTIFIC METHOD IN PHILOSOPHY

Darwin, 14, 22, 33, 41.
Data, 72 ff., 215.

"hard "and "soft," 77 ff.
Dates, 123.
Definition, 209.
Descartes, 15, 80, 242.
Descriptions, 206, 217.
Desire, 231, 237.
Determinism, 237.
Doubt, 240.
Dreams, 93, zoi.
Duration, 153, 157.

Earlier and later, 121.
Eddington, 131 n.
Effect, 224.
Eleatics, 30.
Empiricism, 46, 225.
Enclosure, 120 ff., 127.
Enumeration, 207.
Euclid, 168.
Evellin, 173.
Evolutionism, 14, 21 ff.
Extension, 152, 155.
External world, knowledge of,
70 ff.

Fact, 60.

atomic, 61.
Finalism, 23.

Form, logical, 50 ff., 190, 212.
Fractions, 138, 184.
Free will, 215, 231 ff.
Frege, 15, 50, 2045.

Galileo, 14, 69, 196, 199, 243,

244.

Gaye, 173 n., 180, 182.
Geometry, 15.
Giles, 2ii n.
Greater and less, 199.

Harvard, 14.

Hegel, 13, 47 ff., 56, 173.
" Here," 80, 99-
Hereditary properties, 220.
Hippasos, 168, 241.
Hui Tzft, 21 x.
Hume, 220, 225.
Hypotheses in philosophy,
243.

Illusions, 93.
Incommensurables, 166 ff.,

241.
Independence, 80, 81.

causal and logical, 81, 82.
Indiscernibility, 147, 154.
Indivisibles, 165.
Induction, 43, 225.

mathematical, 199 ff.
Inductiveness, 194, 199 ff.
Inference, 53, 63.
Infinite, 8, 71, 139, 155.

historically considered,

159 &

"true," 184, 185.

positive theory of, 189 ff.
Infinitesimals, 139.
Instants, 122 ff., 135, 153, 219.

defined, 124.

Instinct v. Reason, 30 ff.
Intellect, 32 ff.
Intelligence

how displayed by friends,
101.

inadequacy of display, 103.
Interpretation, 151.

James, 14, 20, 23.
Jourdain, i6gn.
Jowett, 172.
Judgment, 67.

INDEX

249

Kant, 13, 118, 122, 159 ff.,

204.

Keynes, 225 .
Knowledge about, 151.

Language, bad, 89, 142.
Laplace, 22.
Laws of nature, 219 ff
Leibniz, 23, 49, 94, 190, 195.
Logic, 205.

analytic not constructive, 1 8.

Aristotelian, 15.

and fact, 62.

inductive, 43, 225.

mathematical, 8, 49 ff .

mystical, 54.

and philosophy, 18, 42 ff.,

241.
Logical constants, 213.

Mach, 131, 226.
Macran, 48 n.
Mathematics, 49, 66.
Matter, 83, 106 ff.

permanence of, 107 fL
Measurement, 167.
Memory, 233, 237, 239.
Method

deductive, 15.

logical-analytic, 7, 74, 214,

240 ff.

Milhaud, 172 .> 173 n.
Mill, 43, 204.
Montaigne, 39.
Motion, 136, 219.

continuous, 139, 142.

mathematical theory of,

139.

perception of, 143 ff.
Zeno's arguments on, 172 ff.
Mysticiam, 29, 56, 70, 103.

Newton, 41, ^53.
Nicod, 121.
Nietzsche, 20, 21.
Noa, 173.
Number

cardinal, 137, 190 S.

denned, 203 ff.

finite, 165, 193 if.

inductive, 199.

infinite, 183, 186, 190 fL,

201.

reflexive, 1943.

Occam, 112, 153.

One and many, 172, 174.

Order, 137.

Parmenides, 70, 170 ff., 183.
Past and future, 227, 237 ff.
Peano, 50.

Perspectives, 94 ff., 116.
Philoponus, 1761*.
Philosophy

and ethics, 37 ff.

and mathematics, 189 ff.

province of, 27, 36, 189, 236.

scientific, n, 16, 18, 29,

240 ff.
Physics, io6ff., 153, 243, 246.

descriptive, 227.

verifiability of, 88, 116.
Place, 93. 97-

at and front, 100.
Plato, 14, 29, 37* 55. 7 2

1701*., 171.
PoincarS, 131, 148.
Points, H9ff., I35i 1^2.

definition of, 8, 119.
Pragmatism, 21.
Prantl, 179.
Predictability, 232 ff.
Premisses, 214.

250 SCIENTIFIC METHOD IN PHILOSOPHY

Probability, 45.
Propositions! 62.

atomic, 62.

general, 65.

molecular, 64.
Pythagoras, 29, 164 ff., 241.

Race-course, Zeno's argument

of, 175 ff.
Realism, new, 16.
RefLexiveness, 194 ff.
Relations, 54.

asymmetrical, 57.

Bradley's reasons against,
16.

external, 157.

intransitive, 58.

multiple, 60.

one-one, 207.

reality of, 59.

symmetrical, 57, 127.

transitive, 58, 127.
Relativity, 109, 246.
Repetitions, 233 ff.
Rest, 142.

Ritter and Preller, 166 n.
Robertson, D. S., 164 n.
Rousseau, 30.
Royce, 60.

Santayana, 55.
Scepticism, 73, 74.
Seeing double, 93.
Self, 81.
Sensation, 35, 83, 131.

and stimulus, 145.
Sense-data, 63, 70, 72, 82,

116, 148, 150, 216.
and physics, 8, 71, 88, 104,

io6ff., 146.

infinitely numerous ? 156,
163.

Sense-perception, 63.
Series, 59.

compact, 138, 148, 183.

continuous, 138, 139.
Sigwart, 191.
Simplitius, 175 n.
Simultaneity, 121.
Space, 80, 96, 109, 117 ff.,

135-
absolute and relative, 153,

163.

antinomies of, 159 ff .
perception of, 75.
of perspectives, 95 ff.
private, 96* 97-
of touch and sight, 85,

118.

Spencer, 14, 22, 240.
Spinoza, 55, 171.
Stadium, Zeno's argument of,

141 n., 180 ff.
Subject-predicate, 54.
Synthesis, 160, 189.

Tannery, Paul, 173 n.

Teleology, 227.

Testimony, 74, 79, 89, 95,

101, 215.
Thales, 13.

Thing-in-itself, 83, 92.
Things, 96 ff., noff., 216.
Time, 107, 120 ff., 135, 159 ff.,
171, 218.

absolute or relative, 152.

local, 109.

private/ 128.

Uniformities, 220.
Unity, organic, 19.
Universal and particular,
48*.

INDEX

25*

Volition, 227 ff.

Whitehead, 8, 131, 212.
Wittgenstein, g, 213*1.
Worlds

actual and ideal, 116.

Worlds
possible, 190.
private, 95.

Zeller, 178.

Zeno, 135, 140, 142, 169 ff.

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