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Title: Our Knowledge of the External World as a Field for Scientific Method in Philosophy
Author: Russell, Bertrand, 1872-1970
Language: English
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  [ Transcriber's Notes:

    Every effort has been made to replicate this text as faithfully
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  OUR KNOWLEDGE OF THE
  EXTERNAL WORLD

  AS A FIELD FOR SCIENTIFIC METHOD
  IN PHILOSOPHY



BY THE SAME AUTHOR


  INTRODUCTION TO MATHEMATICAL PHILOSOPHY
  Second Edition.
  Demy 8vo, 12s. 6d. net.

  THE ANALYSIS OF MIND
  Demy 8vo, 16s. net.

  PRINCIPLES OF SOCIAL RECONSTRUCTION
  Seventh Impression.
  Cr. 8vo, 5s. net; Limp, 3s. 6d. net.

  ROADS TO FREEDOM: SOCIALISM, ANARCHISM AND SYNDICALISM
  Fourth Impression.
  Cr. 8vo, 5s. net; Limp, 3s. 6d. net.

  THE PRACTICE AND THEORY OF BOLSHEVISM
  Second Impression.
  Cr. 8vo, 6s. net.



  OUR KNOWLEDGE OF
  THE EXTERNAL WORLD
  AS A FIELD FOR SCIENTIFIC METHOD
  IN PHILOSOPHY

  BY
  BERTRAND RUSSELL, F.R.S

  LONDON: GEORGE ALLEN & UNWIN LTD
  RUSKIN HOUSE, 40 MUSEUM STREET, W.C. 1


  First published in 1914 by
  The Open Court Publishing Company

  Reissued by George Allen & Unwin Ltd.
  1922



PREFACE


The following lectures[1] are an attempt to show, by means of examples,
the nature, capacity, and limitations of the logical-analytic method in
philosophy. This method, of which the first complete example is to be
found in the writings of Frege, has gradually, in the course of actual
research, increasingly forced itself upon me as something perfectly
definite, capable of embodiment in maxims, and adequate, in all branches
of philosophy, to yield whatever objective scientific knowledge it is
possible to obtain. Most of the methods hitherto practised have
professed to lead to more ambitious results than any that logical
analysis can claim to reach, but unfortunately these results have always
been such as many competent philosophers considered inadmissible.
Regarded merely as hypotheses and as aids to imagination, the great
systems of the past serve a very useful purpose, and are abundantly
worthy of study. But something different is required if philosophy is to
become a science, and to aim at results independent of the tastes and
temperament of the philosopher who advocates them. In what follows, I
have endeavoured to show, however imperfectly, the way by which I
believe that this _desideratum_ is to be found.

  [1] Delivered as Lowell Lectures in Boston, in March and April 1914.

The central problem by which I have sought to illustrate method is the
problem of the relation between the crude data of sense and the space,
time, and matter 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_.[2] I owe to
him the definition of points, the suggestion for the treatment of
instants and "things," and the whole conception of the world of physics
as a _construction_ 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
Mathematica_.[3] 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.

  [2] London and New York, 1912 ("Home University Library").

  [3] The first volume was published at Cambridge in 1910, the second in
  1912, and the third in 1913.

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 briefly discussed in these lectures, I have had the benefit of
vitally important discoveries, not yet published, 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 modification, 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                                     3

      II. Logic as the Essence of Philosophy                    33

     III. On our Knowledge of the External World                63

      IV. The World of Physics and the World of Sense          101

       V. The Theory of Continuity                             129

      VI. The Problem of Infinity considered Historically      155

     VII. The Positive Theory of Infinity                      185

    VIII. On the Notion of Cause, with Applications to
             the Free-will Problem                             211

          Index                                                243



LECTURE I

CURRENT TENDENCIES


Philosophy, from the earliest times, has made greater claims, 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 philosophers ever since Thales was contradicted by
Anaximander. 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
philosophers have been excessive, and why their achievements have not
been greater. The problems and the method of philosophy have, I believe,
been misconceived 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 distinguish 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 downwards. The
second type, which may be called evolutionism, 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
searching 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 represents, 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 criticise 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 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 has 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 knowledge is literary, 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 groping
where our ancestors walked in the clear daylight of unquestioning
certainty.

The original impulse out of which the classical tradition developed was
the naïve 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 established
with a certainty which no contrary observations could shake. As the
vital impulse of the early philosophers died away, its place was taken
by authority and tradition, reinforced, in the Middle Ages and almost to
our own day, by systematic theology. 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 living 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 Bradley's method. The
world appears to be full of 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, are 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 relation 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 link to connect them with the old.
The problem is to find how the relation can stand to its qualities, and
this problem is insoluble."[4]

  [4] _Appearance and Reality_, pp. 32-33.

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 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 classical tradition. In that tradition, logic becomes
constructive 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 realised 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 experience, it is analytic rather than constructive; taken _a
priori_, it shows the possibility of hitherto unsuspected alternatives
more often than the impossibility of alternatives which seemed _primâ
facie_ 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 paradoxical 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 philosophical 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
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
particularised into the form which states that the relation of knower
and known is fundamental, 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 surviving child of two
very diverse parents: the Greek belief in reason, and the mediæval
belief in the tidiness 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 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 bloodthirstiness 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.


B. Evolutionism

Evolutionism, in one form or another, is the prevailing 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 consonance with the spirit of the
age. It believes itself firmly based on science, a liberator of hopes,
an inspirer of an invigorating faith in human power, a sure antidote to
the ratiocinative authority of the Greeks and the dogmatic authority of
mediæval 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
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. Evolutionism, 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 discipline 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
primitive 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 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 attractiveness 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 the absolute dominion
of change. After explaining why he does not accept mechanism, he
proceeds:[5]

"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 realise a programme
previously arranged. But if there is nothing unforeseen, no invention or
creation in the universe, time is useless again. As in the mechanistic
hypothesis, 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 inflections 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 something 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."

  [5] _Creative Evolution_, English translation, p. 41.

M. Bergson's form of finalism depends upon his conception of life. Life,
in his philosophy, is a continuous 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 carry 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 rendered
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
regarded as really touching any of the questions that to my mind
constitute genuine philosophy.

(1) What biology has rendered probable is that the diverse species arose
by adaptation from a less differentiated 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 millions 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 generalisation 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 philosophy. Evolutionism
thus consists of two parts: one not philosophical, but only a hasty
generalisation 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 evolution 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 philosophers should acquire the disinterested intellectual
curiosity which characterises 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
proposition 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 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
disastrous 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 generalisation 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, 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 reconstruction 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 reconstruction 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 close
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 Ionians were more
scientific and the Sicilians more mystical.[6] 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. 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 the 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.

  [6] _Cf._ Burnet, _Early Greek Philosophy_, pp. 85 ff.

The logic used in defence of mysticism seems to me faulty as logic, and
in a later lecture I shall criticise it on this ground. But the more
thorough-going mystics do not employ logic, which they despise: they
appeal instead directly to the immediate deliverance 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 matters 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
something 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
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
suggested 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 artificial forms of government and thought, and then,
as the purely rationalistic defence of traditional theology 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.
Bergson, 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 harmonising, 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 held with such determination that no
degree of inconsistency with other beliefs leads to their abandonment.
Instinct, like all human faculties, is liable to error. Those in whom
reason is weak are often unwilling 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 careful disguises.
But even in such matters a wrong impression 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 necessitate the
harmonising 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 onesidedness,
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 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_."[7] 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 imperfect medium of words, the knowledge gained by intuition, and
the consequent complete condemnation of all the pretended knowledge
derived from science and common sense.

  [7] _Introduction to Metaphysics_, p. 1.

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 remarkable feats of instinct
in animals, and by pointing out characteristics of the world which,
though intuition can apprehend them, are baffling to intellect as he
interprets it.

Of Bergson's theory that intellect is a purely practical 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 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. Intuition 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 civilised societies men are not
put to death for mathematical 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 intuition and intellect have been
developed because they are useful, and that, speaking broadly, they are
useful when they give truth and become harmful when they give falsehood.
Intellect, in civilised 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
civilisation increases. Speaking broadly, it is greater in children than
in adults, in the uneducated 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 infallibility
as Bergson claims for it. The best 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 think
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 certainly 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 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 intuition 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 civilised men. It is hardly to be supposed,
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 civilised 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, 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 intellect
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 characteristic--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 closely analogous attitudes
towards objects. The general form and structure of those attitudes
towards objects which constitute mental phenomena is a problem for
philosophy; 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 background:
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 found necessary in all the other
sciences. The physicist 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
disinterested 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
_Timæus_ 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 contrary, though he has no wish to deny that
the earth is admirable, is not concerned, as physicist, with its ethical
attributes: 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 particular
sciences, the belief that the notions of good 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 Montaigne, nothing
seems worth while except the discovery of more and more facts, each in
turn the deathblow 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. Thus alongside of despair as 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
possibilities 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
attainment 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 contemplation of truth.

The philosophy, therefore, which is to be genuinely inspired by the
scientific spirit, must deal with somewhat 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 constitution 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
generalisation as to the nature of the universe at large. Many hopes
which inspired philosophers in the past it cannot claim 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 problems 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 authorities as an excellent "propædeutic," _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 century, all
vigorous minds that have concerned themselves with inference have
abandoned the mediæval 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 foundations 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
deduction by pointing out a way of deducing which is certainly not
syllogistic, and does not fit into the mediæval 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. 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: (1) that it is itself known _a priori_; (2) that it is a
postulate; (3) that it is an empirical generalisation 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 unplausible 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
generalisation, which is the view held by Mill.

But if so, how are empirical generalisations 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, "consists in ascribing the nature of general truths to all
propositions which are true in every instance that we happen to know
of."[8] As regards its fallibility, he asserts that "the precariousness
of the method of simple enumeration is in an inverse ratio to the
largeness of the generalisation. 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."[9]

  [8] _Logic_, book iii., chapter iii., § 2.

  [9] Book iii., chapter xxi., § 3.

In the above statement, there are two obvious lacunæ: (1) 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 difficulties in the notion of
probability, but we may ignore them at present. We thus have what at
least _may_ be a logical principle, 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 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[10] 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.

  [10] Or rather a propositional function.

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
empiricists allow, and that it cannot itself be justified
empirically.[11]

  [11] The subject of causality and induction will be discussed again in
  Lecture VIII.

A similar conclusion can be proved by similar arguments concerning any
other logical principle. Thus logical knowledge is not derivable from
experience alone, and the empiricist's philosophy can therefore not be
accepted in its entirety, in spite of its excellence in many matters
which lie 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 conception 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 criticises 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.

The way in which, as it seems to me, Hegel's system assumes the ordinary
logic which it subsequently criticises, 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 form. This belief, being traditional, scarcely
self-conscious, and not supposed to be important, operates underground,
and is assumed in arguments 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 "concrete
universal" and the "union of identity in difference"--will be found
where he explicitly deals with formal logic.[12]

  [12] See the translation by H. S. Macran, _Hegel's Doctrine of Formal
  Logic_, Oxford, 1912. Hegel's argument in this portion of his "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 the hemlock." Owing to this confusion, he
  thinks 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 synthesise
  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 are built upon stupid and trivial
  confusions, which, but for the almost incredible fact that they are
  unintentional, one would be tempted to characterise as puns.

There is quite another direction in which a large technical development
of logic has taken place: I mean the direction of what is called
logistic or mathematical logic. This kind of logic is mathematical in
two different senses: it is itself a branch of mathematics, 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 fulfilment 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 realising
that this was possible.[13]

  [13] _Cf._ Couturat, _La Logique de Leibniz_, pp. 361, 386.

The modern development of mathematical logic 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 mathematics. Traditional
logic regarded the two propositions, "Socrates is mortal" and "All men
are mortal," as being of the same form;[14] 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.

  [14] It was often recognised that there was _some_ difference between
  them, but it was not recognised that the difference is fundamental,
  and of very great importance.

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 properly be called
_philosophical_ logic, I shall speak shortly. But even the later
developments, though not directly philosophical, will be found of great
indirect use in philosophising. They enable us to deal easily with more
abstract conceptions than merely verbal reasoning can enumerate; they
suggest fruitful hypotheses 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
abstraction," and is one which clears away incredible accumulations 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 similar 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 time to turn our attention to its philosophical foundations.

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
propositions 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
constituent 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 are 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 close 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
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 form without having
knowledge of the constituents. If I say, "Rorarius drank the hemlock,"
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 both of the constituents and of the particular instance of the
form. It is in this way that a sentence conveys information, 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 particular 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 are mentioned: the
proposition is absolutely general. All inferences, when stated fully,
are instances of propositions having this kind of 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 are verified and when they are
not.

But the forms of propositions giving rise to inferences are not the
simplest forms: they are always hypothetical, stating that if one
proposition is true, then so is another. Before considering inference,
therefore, logic must consider those simpler forms which inference
presupposes. 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." Grammar
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
propositions stating that two things have a certain relation have a
different form from subject-predicate propositions, 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 propositions 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 feels 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 independent of the sudden illumination from which they
sprang. Nevertheless their origin clung 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 can account for
the complacency with which philosophers have accepted the inconsistency
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 sister of B, then 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 non-symmetrical, 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 asymmetrical 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--for 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, 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 expressing
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
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 predicates 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 _logical_ 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 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.[15] 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.

  [15] _Encyclopædia of the Philosophical Sciences_, vol. i. p. 97.

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 between which 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 constituents. When it simply assigns a quality to
a thing, 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 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 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-perception 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.[16] 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 wholly 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.

  [16] 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.

"Molecular" propositions are such as contain conjunctions--_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 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 form
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
philosophers 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 manner.

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 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 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 it 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, theoretically, 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 two terms. The
case of judgment demands the admission of more complicated forms. If all
judgments were true, we might suppose that a judgment consisted in
apprehension of a _fact_, 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 apprehension. 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 realise this necessity has, in my opinion, vitiated almost
everything that has hitherto been written on the theory of knowledge,
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 _primâ facie_ 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 _primâ facie_ hypotheses as a rule remain
admissible, others, which only logic would have suggested, are 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 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 appearance criticised 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
organisation and point of view of the spectator; while modern physics,
on the basis of sensible evidence itself, maintains a mad dance of
electrons which has, superficially 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_ 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 Parmenides 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 immediate 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 knowledge 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 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 interpreted in terms of sense.

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 case of our present problem, the common knowledge
involved is of various kinds. There is first our acquaintance with
particular objects of daily life--furniture, houses, towns, other
people, and so on. Then there is the extension of such particular
knowledge to particular things outside our personal experience, through
history and geography, newspapers, etc. And lastly, there is the
systematisation of all this knowledge of particulars by means of
physical science, which derives immense 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 knowledge, 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 philosophical analysis.

It may be said--and this is an objection which must be met at the
outset--that it is the duty of the philosopher 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 criticise the whole of the knowledge of daily life. 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
boast of having achieved such a degree of certainty that it can have
authority to condemn the facts of experience and the laws of science.
The philosophic scrutiny, therefore, though 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 abstention 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 criticised 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 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 speculations 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 instinctively 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 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 realised, the exact
outcome does not make any very great difference in our main problem.

The next step in our analysis must be the consideration of how the
derivative parts of our common knowledge arise. Here we become involved
in a somewhat 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
constantly 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 separation of these two kinds of primitiveness is
vitally important to our present discussion.

When we reflect upon the beliefs which are 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
psychologically 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 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 realise 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 philosophising, 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 primitive 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 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
sometimes 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 remember that the distinction
of hard and soft data is psychological 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
obviously derivative from our perception of their bodies, and is felt to
demand logical justification as soon as we become aware of its
derivativeness. 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 recondite 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 feel 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. 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
"independent" 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, (1) 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 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 perceiving them?
Secondly, if this cannot be known, can we know that other objects,
inferable from objects 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.

Owing to the fact that we feel passive in sensation, we naturally
suppose that our sensations have outside causes. Now it is necessary
here first of all to distinguish between (1) our sensation, which is a
mental event consisting in our being aware of a sensible object, and (2)
the sensible object of which we are aware in sensation. When I speak of
the 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. Each of
these I call a sensible object, and our awareness of it I call a
sensation. Now our sense of passivity, if it really afforded any
argument, would only tend to show that the _sensation_ has an outside
cause; this cause we should naturally seek in the sensible object. Thus
there is no good reason, so far, for supposing that sensible objects
must have outside causes. But both the thing-in-itself of philosophy and
the matter of physics present themselves as outside causes of the
sensible object as much as of the sensation. What are the grounds for
this common 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
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. The frame of the spectacles is of
course visible, but the blue glass, if it is clean, 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 completely 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 appearance 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 assumption is actually
unavoidable, though it is unquestionably the most natural one to make.
We may say that the 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 prejudice, 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 feel 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, 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 sensations, 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 transparent 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 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 immediately
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 close 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 belong 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
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 verification 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 summarise 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 "independent" precisely. The felt passivity of
sensation is irrelevant, since, even if it proved anything, it could
only prove that sensations are caused by sensible objects. The natural
_naïve_ 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 discovered 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 experience, 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 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 are 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.
There is, in this theory, a tendency to a confusion from which it
derives some of its plausibility, namely, the confusion between a
sensation as a psychical occurrence and its object. A patch of colour,
even if it only exists when it is seen, is still something quite
different from the seeing of it: the seeing of it is mental, but the
patch of colour is not. This confusion, however, can be avoided without
our necessarily abandoning the theory we are examining. The objection to
it, I think, lies in its failure to realise 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 clear.
But what remains far from clear 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 nevertheless 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 susceptible of a true
interpretation. Let us see how this is to be done.

The first thing to realise is that there are no such 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! 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 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-dimensional world; but there is absolutely
nothing which is seen by two minds simultaneously. When we say 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 perspectives.

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 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
common-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 mere logical construction. It has, however, the
merit of being 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 are the
elements. There are as many private spaces as there are perspectives;
there are therefore at least as many as there are percipients, 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
correlated 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 perspective in question, is circular,
not elliptic. We can then form a whole series of perspectives containing
a graduated 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. Moreover--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 perspectives 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 perspectives
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 spatial 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
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 thing looks large are nearer to the thing 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 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: (1) 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 experienced facts, including those derived
from testimony. 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 impossibilities. 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 conform 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 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 regarding 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 connecting 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!" 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 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
convincingness that belongs to our morning realisation 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 suppose 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 systematises 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 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 appearance 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
hypothetical 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 hypothetical
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 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 difference 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_ principle 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 psychologists, who are conscious of
it, have not the mathematical 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.

Let us begin by a brief description of the two contrasted 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 permanent and fairly rigid bodies--tables
and chairs, stones, mountains, the earth and moon and sun. This
common-sense belief, it should be noticed, is a piece of audacious
metaphysical theorising; 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 departure 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 are 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, 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 is developing 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 pictorial accounts which physicists give of the material world as
they conceive it undergo violent changes under the influence of
modifications in theory which are much slighter than the layman might
suppose from the alterations of the description. Certain features,
however, have remained fairly stable. It is always assumed that there is
_something_ indestructible which is capable of motion in space; what is
indestructible is always very small, but does not always occupy a mere
point in space. There is supposed to be one all-embracing space in which
the motion takes place, and until lately we might have assumed one
all-embracing time also. But the principle of relativity has given
prominence to the conception of "local time," and has somewhat
diminished men's confidence in the one even-flowing stream of time.
Without dogmatising as to the ultimate outcome of the principle of
relativity, however, we may safely say, I think, that it does not
destroy the possibility of correlating different local times, and does
not therefore have such far-reaching philosophical consequences as is
sometimes supposed. In fact, in spite of difficulties as to measurement,
the one all-embracing time still, I think, underlies all that physics
has to say about motion. We thus have still in physics, as we had in
Newton's time, a set of indestructible entities which may be called
particles, moving relatively to each other in a single space and a
single time.

The world of immediate data is quite different from this. Nothing is
permanent; even the things that we think are 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 give relations that may be called spatial.
Experience teaches us to obtain one space from these by correlation, and
experience, together with instinctive theorising, teaches us to
correlate our spaces with those which we believe to exist in the
sensible worlds 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. Thus, apart from any of the fluctuating
hypotheses of physics, three main problems arise in connecting the world
of physics with the world of sense, namely (1) the construction of
permanent "things," (2) the construction of a single space, and (3) the
construction of a single time. We will consider these three problems in
succession.

(1) 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 unchanging. 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 something 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 formulated 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 reconstructing the conception of matter without the _a priori_
beliefs which historically gave rise to it. In spite of the
revolutionary results of modern physics, 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. 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 imperceptible 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 feel 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
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 continuity 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 _of_ 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 philosophising, 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 philosophy, 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-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
different things may have any degree of likeness up to exact similarity.

Another insufficient criterion of one thing is _continuity_. As we have
already seen, if we watch what we regard as one changing thing, we
usually find its changes to be continuous so far 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 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 uninterrupted observation is as a rule a mark of
difference between things, though even this cannot be said in such cases
as sudden explosions.

The assumption of continuity is, however, successfully 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 is
inconsistent 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 may
be allowed to be a _necessary_ condition if two appearances are to be
classed as appearances of the same thing. But it is not a _sufficient_
condition, as appears from the instance of the drops in the sea. Thus
something more must be sought before we can give even the roughest
definition of a "thing."

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 simultaneous
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 unverifiable 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 belonging 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 appearances 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 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 considerations, 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 unverifiable: there are
hypotheses as to (α) how things would appear to a spectator in a place
where, as it happens, there is no spectator; (β) how things would appear
at times when, in fact, they are not appearing to anyone; (γ) 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 to show how
this might possibly be done, let us recall the hypothetical Leibnizian
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 enunciation 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
whenever we choose to look at it. It is open to us to believe that the
ideal elements exist, and there can be no reason for _dis_believing
this; but unless in virtue of some _a priori_ law we cannot _know_ it,
for empirical knowledge is confined to what we actually observe.

(2) The three main conceptions of physics are space, time, and matter.
Some of the problems raised by the conception of matter have been
indicated in the above discussion of "things." But space and time also
raise difficult problems of much the same kind, namely, difficulties in
reducing the haphazard untidy world of immediate sensation to the smooth
orderly world of geometry and kinematics. Let us begin with the
consideration of space.

People who have never read any psychology seldom realise 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 feel 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, 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
customary to think of points as simple and infinitely small, but
geometry in no way demands that we should think 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.

The following illustrative method, simplified so as to be easily
manipulated, has been invented by Dr Whitehead for the purpose of
showing how points might be manufactured from sense-data. We have first
of all to observe that there are no infinitesimal sense-data: any
surface we can see, for example, must be of some finite extent. But what
at first 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 object 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 class of spatial objects, namely all those (as it will turn out
in the end) which would naturally be said to contain the point. In order
to obtain a definition of a "point" in this way, we proceed as follows:

Given any set of volumes or surfaces, they will not in general converge
into one point. But if they get smaller and smaller, while of any two of
the set there is always one that encloses the other, then we begin to
have the kind of conditions which would enable us to treat them as
having a point for their limit. The hypotheses required for the relation
of enclosure are that (1) it must be transitive; (2) of two _different_
spatial objects, it is impossible for each to enclose the other, but a
single spatial object always encloses itself; (3) any set of spatial
objects such that there is at least one spatial object enclosed by them
all has a lower limit or minimum, _i.e._ an object enclosed by all of
them and enclosing all objects which are enclosed by all of them; (4) to
prevent trivial exceptions, we must add that there are to be instances
of enclosure, _i.e._ there are really to be objects of which one
encloses the other. When an enclosure-relation has these properties, we
will call it a "point-producer." Given any relation of enclosure, we
will call a set of objects an "enclosure-series" if, of any two of them,
one is contained in the other. We require a condition which shall secure
that an enclosure-series converges to a point, and this is obtained as
follows: Let our enclosure-series be such that, given any other
enclosure-series of which there are members enclosed in any arbitrarily
chosen member of our first series, then there are members of our first
series enclosed in any arbitrarily chosen member of our second series.
In this case, our first enclosure-series may be called a "punctual
enclosure-series." Then a "point" is all the objects which enclose
members of a given punctual enclosure-series. In order to ensure
infinite divisibility, we require one further property to be added to
those defining point-producers, namely that any object which encloses
itself also encloses an object other than itself. The "points" generated
by point-producers with this property will be found to be such as
geometry requires.

(3) 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 clearly 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 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. At 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
itself, 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, are all that experience provides. Hence, unless we are
to introduce superfluous metaphysical entities, we must, in defining
what mathematical physics can regard as an instant, 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
simultaneous 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 simultaneous 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 simultaneous 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
way by which a completely accurate date can be defined.

        A____________________

    B____________________

            C________

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 simultaneous 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 simultaneous with, some event in
the group which is the 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 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 precedes 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" 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 "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.[17]

  [17] The assumptions made concerning time-relations in the above are
  as follows:--

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

      (a) No event wholly precedes itself. (An "event" is defined as
      whatever is simultaneous with something or other.)

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

      (c) 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:

      (f) 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._ N.
  Wilner, "A Contribution to the Theory of Relative Position," _Proc.
  Camb. Phil. Soc._, xvii. 5, pp. 441-449.

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

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 be a "point-producer," we require (1) that it
should be transitive, _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 divisibility, we require also that every event should enclose
events other than itself. Assuming these characteristics, temporal
enclosure is an infinitely divisible point-producer. We can now form an
"enclosure-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.

The correlation of the times of different private worlds so as to
produce the one all-embracing time of physics is a more difficult
matter. We saw, in Lecture III., that different private worlds often
contain correlated 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; but from the
point of view of philosophical theory, there is no very serious
difficulty of principle involved.

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 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 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 throughout
the civilised world. Physicists, ignorant and contemptuous 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. Metaphysicians,
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 therefore been content to say
that matter, space, and time are "intellectual constructions," 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
recognise 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.

There are, it is true, two authors, both physicists, who have done
something, though not much, to bring about a recognition of the problem
as one demanding study. These two authors are Poincaré and Mach,
Poincaré especially in his _Science and Hypothesis_, 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.
Poincaré is Kantian, while Mach is ultra-empiricist; with Poincaré
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 "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 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 simultaneity, _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 follow 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 transitive 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 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 transitive 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 ordinarily
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 speaking, a part of philosophy. 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
philosophy is, broadly speaking, the following: Space and time are
treated by mathematicians 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 and instants.

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 continuity, 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 realise 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 mathematical doctrine of continuity as an
inadequate explanation of the continuity which we experience in the
world of sense.

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 mathematicians 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 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 also be arranged in orders
which are not continuous. Thus the essence of continuity 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 "continuous," for technical purposes, to series
having a certain high degree of continuity. But for philosophical
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 1/2:
if we choose some fraction which is very little greater than 1/2, say
51/100 we can find others, such as 101/200, which are nearer to 1/2.
Thus between any two fractions, however little they differ, there are an
infinite number of other fractions. Mathematical space and time also
have this property of compactness, though whether actual space and time
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 realise 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,
compactness 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 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 distance, 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 fall into error, because there is
no _next_ point or _next_ instant. If there were, we should find Zeno's
paradoxes, in some form, unavoidable, 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 hypothesis upon which it is based, namely that there
are consecutive points and instants.[18] Hence the continuity of motion
must not be supposed to consist in a body's occupying consecutive
positions at consecutive times.

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

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 continue 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 our original distance was an inch, we reach
successively half an inch, a quarter of an inch, an eighth, a sixteenth,
and so on; but every one of this infinite series of diminishing
distances is finite. "But," it may be said, "_in the end_ the distance
will grow infinitesimal." 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 distance, we can find a smaller distance; this
may be expressed in the ambiguous form "there is a distance smaller than
any finite distance." But if this is then 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 position, 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 together we 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;[19] 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.

  [19] See next lecture.

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 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 P1P2 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 _t_, it must be possible to find two
instants _t_1, _t_2, one earlier than _t_ and one later, such that
throughout the whole time from _t_1 to _t_2 (both included), the
particle lies between P1 and P2. And we say that this must still hold
however small we make the portion P1P2. 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 P1P2 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.

        P1   P    P2   Q
  ------|----|----|----|------>

Philosophers, mostly in ignorance of the mathematician's analysis, have
adopted other and more heroic methods of dealing with the _primâ facie_
difficulties 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.[20]

  [20] _Monist_, July 1912, pp. 337-341.

Apart from definite arguments, there are certain feelings, rather than
reasons, which stand in the way of an acceptance of the mathematical
account of 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 something
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 mathematical
theory of motion is required to account for it. Opponents of the
mathematical theory emphasise 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 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 mathematician.

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

(1) The physiological answer merely shows that, if the physical world is
what the mathematician 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
fainter; thus 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, until 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 showing 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 application 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 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
showing 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, 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 propositions 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_.

When we are considering the actual data of sensation in this connection,
it is important to realise 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
emphasised by Poincaré.[21] 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.

  [21] "Le continu mathématique," _Revue de Métaphysique et de Morale_,
  vol. i. p. 29.

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 reasonable to suppose that sense-data of a given kind,
such as weights or colours, really form a compact series. The objections
which may be brought from a psychological 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 formulæ; and in
this sense the mathematical theory of motion is applicable to the data
of sensation as well as to the supposed particles of abstract physics.

There are a number of distinct questions which are apt to be confused
when the mathematical continuum 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 continuity logically
    possible?

    (b) 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?

    (d) 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 elementary misunderstandings we considered
at the beginning of the present lecture, partly on the possibility 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 complete the psychological
answer.

(b) 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 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 ad
absurdum_ 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." Acquaintance, which is what we derive from sense,
does not, theoretically at least, imply even the smallest "knowledge
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
acquaintance 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 nonexistent. 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 propositions, which is not involved
necessarily in acquaintance with the constituents of the propositions.
To know that two shades of colour are different is knowledge 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 that they are not composed of mutually
external units. It may be admitted, on the other hand, that nothing in
their empirical character specially necessitates 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 are adequate to the conclusion. They rest, at bottom, upon the
impossibility of explaining 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 assumption 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
questions in turn may take two forms, namely: (α) is the hypothesis
_consistent_ with the facts and with logic? (β) 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,
_i.e._ it assumes that, besides the things which are in space and time,
there are also entities, called "points" and "instants," which are
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 temporal
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-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.

(d) 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
hypothesis. 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 discrimination. 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 empirically
discoverable to show that objects of sense are not continuous. In what
is called _experienced_ continuity, 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 indistinguishability 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 elements 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 hypotheses 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 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 considerations 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 know whether 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. I do not know what is the right course to adopt as regards
these alternatives. The logical analysis we have been considering
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 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
concerning any one thing. I do not think the philosophers in question
have realised that this is the precise statement 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 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 impossible 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 made by a suitable
interpretation. The direct consideration of the crude facts of the
physical world has been undertaken 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 contradictions
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
regards space is enclosed within limits"; the antithesis 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 synthesis. 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 minimum. 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 _classes_, and only
derivatively applicable to series; classes which are infinite are given
all at once by the defining property of their members, so that there is
no question of "completion" or of "successive synthesis." And the word
"synthesis," by suggesting the mental activity of synthesising,
introduces, more 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 backwards 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 him to attribute validity
to a singularly flimsy piece of fallacious reasoning.

The second antinomy illustrates the dependence of the problem of
continuity upon that of infinity. The thesis states: "Every complex
substance in the world consists of simple parts, and there exists
everywhere 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 criticism,
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) consists 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 consists 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 proposition
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.
Geometry 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 _primâ 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 think it impossible that
space should be composed of points?

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
divisibility. A space may be halved, and then halved again, and so on
_ad infinitum_, and at every stage of the process 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 denying 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
lecture, I wish to state and explain the problem of infinity, to show
how it arose, and to show the irrelevance 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
make this problem peculiarly apt for the illustration of method.

The problem appears to have first arisen in some such way as the
following.[22] Pythagoras and his followers, who were interested, like
Descartes, in the application of 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 composed of indivisible points, while time is
composed of indivisible instants.[23] This belief would not, by itself,
have raised the difficulties which they encountered, 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 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 0 and 1 and
2 and 3 and so on, 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.

  [22] 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.

  [23] _Cf._ Aristotle, _Metaphysics_, M. 6, 1080b, 18 _sqq._, and
  1083b, 8 _sqq._

Whether the Pythagoreans themselves believed space and time to be
composed of indivisible points and instants is a debatable question.[24]
It would seem that the distinction between space and matter had 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[25] in
Aristotle's _Physics_,[26] 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
consecutives, 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."

  [24] There is some reason to think that the Pythagoreans distinguished
  between discrete and continuous quantity. G. J. Allman, in his _Greek
  Geometry from Thales to Euclid_, says (p. 23): "The Pythagoreans made
  a fourfold division of mathematical science, attributing one of its
  parts to the how many, τὸ πόσον, and the other to the how much, τὸ
  πηλίκον; 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 (τὴν σφαιρικήν) contemplates continued
  quantity so far as it is of a self-motive nature. (Proclus, ed.
  Friedlein, p. 35. As to the distinction between τὸ πηλίκον,
  continuous, and τὸ πόσον, discrete quantity, see Iambl., _in Nicomachi
  Geraseni Arithmeticam introductionem_, ed. Tennulius, p. 148.)" _Cf._
  p. 48.

  [25] Referred to by Burnet, _op. cit._, p. 120.

  [26] iv., 6. 213b, 22; H. Ritter and L. Preller, _Historia Philosophiæ
  Græcæ_, 8th ed., Gotha, 1898, p. 75 (this work will be referred to in
  future as "R. P.").

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 arithmetical 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 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.[27] Thus the length of the side and the length of the diagonal
are incommensurable; 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_.

  [27] 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
  _m_2 = 2_n_2. Now the square of an odd number is odd, but _m_2, being
  equal to 2_n_2, is even. Hence _m_ must be even. But the square of an
  even number divides by 4, therefore _n_2, which is half of _m_2, must
  be even. Therefore _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.

Now this fact might have been assimilated by some 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 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
Metapontion, is even said to have been shipwrecked at sea for impiously
disclosing the terrible discovery to their enemies. It must be
remembered that Pythagoras 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 incommensurables 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 arithmetic,
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 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 contradictions. A long line of
philosophers, from Zeno[28] to M. Bergson, have based much of their
metaphysics upon the supposed impossibility of infinite collections.
Broadly speaking, the difficulties were stated by Zeno, and nothing
material was added until we reach Bolzano's _Paradoxien des
Unendlichen_, 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 of the difficulties is
due, not to Bolzano, but to Georg Cantor, whose work on this subject
first appeared in 1882.

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

In order to understand Zeno, and to realise how little modern orthodox
metaphysics has added to the achievements of the Greeks, we must
consider for a moment his master Parmenides, in whose interest the
paradoxes were invented.[29] 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
Parmenides tells us first about reality and then about appearance. "The
way of opinion," in his philosophy, is, broadly speaking,
Pythagoreanism; 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 principle of his inquiry is stated in a sentence which would
not be out of place in Hegel:[30] "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; for what is past can be spoken of, and therefore,
by the principle, still is.

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

  [30] "With Parmenides," Hegel says, "philosophising proper began."
  _Werke_ (edition of 1840), vol. xiii. p. 274.

The great conception of a reality behind the passing illusions of sense,
a reality one, indivisible, and unchanging, 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 feel than to state--in
which time is an unimportant and superficial 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 realise 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[31] in the
dialogue in which Socrates, as a young man, learns logical acumen and
philosophic disinterestedness 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."

  [31] _Parmenides_, 128 A-D.

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.[32] Unfortunately, we only know his arguments through
Aristotle,[33] who stated them in order to refute them. Those
philosophers in the present day who have had their doctrines stated by
opponents will realise 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.

  [32] This interpretation is combated by Milhaud, _Les
  philosophes-géomètres de la Grèce_, 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.

  [33] _Physics_, vi. 9. 2396 (R.P. 136-139).

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 maintain that they were aimed at the Pythagoreans,[34] while others
have held that they were intended to refute the atomists.[35] M.
Evellin, on the contrary, holds that they constitute a refutation of
infinite divisibility,[36] while M. G. Noël, in the interests of Hegel,
maintains that the first two arguments refute infinite divisibility,
while the next two refute indivisibles.[37] Amid such a bewildering
variety of interpretations, we can at least not complain of any
restrictions on our liberty of choice.

  [34] _Cf._ Gaston Milhaud, _Les philosophes-géomètres de la Grèce_,
  p. 140 n.; Paul Tannery, _Pour l'histoire de la science hellène_,
  p. 249; Burnet, _op. cit._, p. 362.

  [35] _Cf._ R. K. Gaye, "On Aristotle, _Physics_, Z ix." _Journal of
  Philology_, vol. xxxi., esp. p. 111. Also Moritz Cantor, _Vorlesungen
  über Geschichte der Mathematik_, 1st ed., vol. i., 1880, p. 168, who,
  however, subsequently adopted Paul Tannery's opinion, _Vorlesungen_,
  3rd ed. (vol. i. p. 200).

  [36] "Le mouvement et les partisans des indivisibles," _Revue de
  Métaphysique et de Morale_, vol. i. pp. 382-395.

  [37] "Le mouvement et les arguments de Zénon d'Élée," _Revue de
  Métaphysique et de Morale_, vol. i. pp. 107-125.

The historical questions raised by the above-mentioned discussions are
no doubt largely insoluble, owing to the very scanty material from which
our evidence is derived. The points which seem fairly clear are the
following: (1) 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;[38]
(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.

  [38] _Cf._ M. Brochard, "Les prétendus sophismes de Zénon d'Élée,"
  _Revue de Métaphysique et de Morale_, vol. i. pp. 209-215.

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 or 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, 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 will
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."[39]

  [39] Simplicius, _Phys._, 140, 28 D (R.P. 133); Burnet, _op. cit._,
  pp. 364-365.

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
are, they will be finite in number." This phrase is not very clear, 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 impossible.
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 the race-course, which
is paraphrased by Burnet as follows:[40]

"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 distance 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."[41]

  [40] _Op. cit._, p. 367.

  [41] 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. 939B (R.P.
  136). Aristotle seems to refer to _Phys._, vi. 2. 223AB [R.P. 136A]:
  "All space is continuous, for time and space are divided into the same
  and equal divisions.... Wherefore also Zeno's argument is fallacious,
  that it is impossible 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. 136A, _Exc.
  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."

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 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. (1)
If _all_ 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:[42] "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 interpretation, 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 force of the argument, on this
interpretation, lies solely in the mistaken supposition that there
cannot be anything beyond the whole of an infinite series, which can be
seen to be false by observing that 1 is beyond the whole of the infinite
series 1/2, 3/4, 7/8, 15/16, ...

  [42] _Cf._ Mr C. D. Broad, "Note on Achilles and the Tortoise,"
  _Mind_, N.S., vol. xxii. pp. 318-9.

The second of Zeno's arguments is the one concerning Achilles and the
tortoise, which has achieved more notoriety than the others. It is
paraphrased by Burnet as follows:[43]

"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."[44]

  [43] _Op. cit._

  [44] 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. 239B (R.P. 137).

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,[45] that of the arrow, is very interesting. The text
has been questioned. Burnet accepts the alterations of Zeller, and
paraphrases thus:

"The arrow in flight is at rest. For, if everything 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."

  [45] _Phys._, vi. 9. 239B (R.P. 138).

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 Burnet's paraphrase.

Here, if not in the first two arguments, the view that a finite part of
time consists of a finite series of successive 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 position and no next moment, and
when once this is imaginatively realised, the difficulty is seen to
disappear.

The fourth and last of Zeno's arguments is[46] the argument of the
stadium.

  [46] _Phys._, vi. 9. 239B (R.P. 139).

The argument as stated by Burnet is as follows:

  First Position.  Second Position.
      A ....           A  ....
      B ....           B ....
      C ....           C   ....

"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
are 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 half."

Gaye[47] devoted an interesting article to the interpretation of this
argument. His translation of Aristotle'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 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, B B ... 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 C C ... those originally occupying the other
half from the goal to the middle of the A's, equal in number, size, and
velocity, to B B ... 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 consequently 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 reach 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, because 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."

  [47] _Loc. cit._

  First Position.     Second Position.
  B  B′ B″              B  B′ B″
  ·  ·  ·               ·  ·  ·

  A  A′ A″           A  A′ A″
  ·  ·  ·            ·  ·  ·

  C  C′ C″        C  C′ C″
  ·  ·  ·         ·  ·  ·

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 suppose three
drill-sergeants, A, A′, and A″, standing in a row, while the two files
of soldiers march past them in 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 consecutive, 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
difficulty raised by Zeno. What Zeno professes to prove is that "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.[48] 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
contention. If we suppose that time consists of a series of 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 instant. 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.

  [48] _Loc. cit._, p. 105.

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
opposite assumption, 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 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 difficulties can also be met if infinite numbers are
admissible. 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 1, 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. 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
Parmenides, the true philosopher must learn to leave behind him.

The confusions introduced into the notions of philosophers by the
so-called "true" infinite are curious. They see that this notion is not
the same as the mathematical infinite, but they choose to believe that
it is the notion which the mathematicians are vainly trying to reach.
They therefore inform the mathematicians, 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
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 0, has another number
immediately before it, from which it results by adding 1; 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-contradictory. 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 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
collections; 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 difficulty of the subject lies in the necessity
of thinking in an unfamiliar way, and in realising that many properties
which we have thought inherent in number 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 cling obstinately to the prejudices instilled by the
arithmetic which is learnt in childhood.



LECTURE VII

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 philosophical
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-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 philosophy 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 generalise 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 these facts
to others, simpler, more fundamental, and inherently more fitted to form
the premisses of the science of arithmetic. The 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
plurality thought _as held together and closed, and to that extent as a
unity_." 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.

Such vague philosophic definitions, however, remained 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 are 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 attention. 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 one, and those who
imagine that it is the logical source of number show themselves
remarkably incapable 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 realised; but
those who wish to be logicians must acquire the habit of dwelling upon
such facts. There are two propositions involved in this fact: first,
that the number of numbers from 1 up to any given number is that given
number--for instance, the number of numbers from 1 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 concerned; 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 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 collection, 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.

(1) _Reflexiveness._--A number is said to be _reflexive_ when it is not
increased by adding 1 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 recognised 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 0, 1, 2, 3, ... to be written down in a
row, and immediately beneath them write down the numbers 1, 2, 3,
4, ..., so that 1 is under 0, 2 is under 1, 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 0; thus the
number of terms 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 1 to it, this state of things constituted a
contradiction, and led to the denial that there are infinite numbers.

  0, 1, 2, 3, ... _n_ ...
  1, 2, 3, 4, ... _n_ + 1 ...

The following example is even more surprising. Write the natural numbers
1, 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 maintained 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 than
the number of even numbers, _i.e._ the whole is not greater than its
part."[49] In dealing with this 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 maintain 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 realisation of this fact which has made the modern theory of
infinity possible.

  [49] _Phil. Werke_, Gerhardt's edition, vol. i. p. 338.

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 1730.[50] 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.

"_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
_Simplicius_, 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 call'd
Squares, the Factors are call'd 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.

"_Salv._ 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. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, which
is the same as to say the 10th Part are Squares; in Ten thousand only
the 100th Part are Squares; in a Million only the 1000th: And yet in an
infinite Number, if we can but comprehend 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."

  [50] _Mathematical Discourses concerning two new sciences relating to
  mechanics and local motion, 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 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 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 contradictions 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 considerations alone. But this question has not the
fundamental importance which belongs to the arithmetical 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 characterises the finite numbers, 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 0 and 1, between 1 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 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 0 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 0 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 "inductive" properties. Thus an
inductive property of numbers is one which is hereditary and belongs to
0.

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
0 and are hereditary, they belong to 1; 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 that possess all 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 0 by successive
additions of 1; 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 properties. Such numbers,
therefore, may be called non-inductive. All those properties of numbers
which are proved by an imaginary 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 numbers. 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
realise 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 1--_i.e._ the
property of non-reflexiveness--may serve to illustrate the limitations
of mathematical induction. It is easy to prove that 0 is increased by
the addition of 1, and that, if a given number is increased by the
addition of 1, so is the next number, _i.e._ the number obtained by the
addition of 1. It follows that each of the natural numbers is increased
by the addition of 1. 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 0 is not equal to 1;
then, since the property of being increased by 1 is hereditary, it
follows that 1 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 30,001, we can do
so by repeating this reasoning 30,000 times. But we cannot prove in this
way that _all_ numbers are increased by the addition of 1; we can only
prove that this holds of the numbers attainable by successive additions
of 1 starting from 0. The reflexive numbers, which lie beyond all those
attainable in this way, are as a matter of fact not increased by the
addition of 1.

The two properties of reflexiveness and non-inductiveness, 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 arithmetic, 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,
the arithmetical as opposed to the logical part of the theory--was
discovered by Georg Cantor, and published by him in 1882-3.[51] 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, _Begriffsschrift_, 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 contained in his second work, published in 1884,
and entitled _Die Grundlagen der Arithmetik, eine logisch-mathematische
Untersuchung über den Begriff der Zahl_.[52] It is with this book that
the logical theory of arithmetic begins, and it will repay us to
consider Frege's analysis in some detail.

  [51] In his _Grundlagen einer allgemeinen Mannichfaltigkeitslehre_ and
  in articles in _Acta Mathematica_, vol. ii.

  [52] The definition of number contained in this book, and elaborated
  in the _Grundgesetze 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.

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

The fact is, as Frege points out, that no number, not even 1, is
applicable to physical things, but only to general terms or
descriptions, such as "man," "satellite 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 necessary 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, 0 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 0. This was impossible if
numbers applied to physical objects, because obviously no physical
object could have the number 0. 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 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 are 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 class, enumeration is impossible, so that description by a
general characteristic common and peculiar to the members of the class
is the only possible description. Here, as we see, the theory of number
to which Frege was led by purely 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 help to make the method clear. 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 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
question: 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 are defined, of which numbers can be
asserted; and it applies to 0 and 1 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 unfamiliarity 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 formulæ of
arithmetic; any indubitable 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 falsely) about classes of things.
That is to say, if, in any sentence in which a thing is mentioned, 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
meaningless collection of words. Appearances to the contrary can be
dispelled by a moment's reflection. For 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 individuals 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
class 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 Tzŭ, 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."[53] The author
from whom I quote says that Hui Tzŭ "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 irrefragable. 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.

  [53] Giles, _The Civilisation of China_ (Home University Library),
  p. 147.

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 class of people interested in 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_, 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 definition 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.[54]

  [54] Cf. _Principia Mathematica_, § 20, and Introduction, chapter iii.

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 corresponding 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 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.[55] 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.

  [55] In the above remarks I am making use of unpublished work by my
  friend Ludwig Wittgenstein.



LECTURE VIII

ON THE NOTION OF CAUSE, WITH APPLICATIONS 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 knowledge from
them belongs to mathematics, if "mathematics" is interpreted in a
somewhat liberal sense.

But besides the logical analysis of the common knowledge which forms our
data, there is the consideration 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," 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; 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 proposition in virtue of which
it is possible to infer the existence 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 nevertheless 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.[56] 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 existence must be applied to what, at the moment, is not a
datum; it is in the possibility of such application 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.

  [56] 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.

But we cannot become acquainted with a particular except by its being
actually given. Hence the particular inferred by a causal law 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 claps 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.

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

Often a causal law involves 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 lead to a
belief in causal laws, not whether these grounds are adequate to support
the belief in universal causation.

The first step is the discovery of approximate unanalysed 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 example 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 psychological account
of what may be called the animal belief in causation, because it is
something 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
sometimes see lightning without hearing thunder; and although, in such a
case, we suppose that thunder 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 uniformity. Unsupported bodies in air fall,
unless they are balloons or aeroplanes; but the principles of mechanics
give uniformities which apply to balloons and aeroplanes 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
circumstances 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 doubtful, 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
instant an acceleration compounded of accelerations towards all the
other bodies in the solar system, proportional to the masses of those
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 are 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. Exceptions 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 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
generalisations 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
realised 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 circumlocution.

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 different 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 sequence, 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 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, 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 thing 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 approaches 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 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 emphasised
the scope of induction wished to maintain that all logic is empirical,
and therefore could not be expected to realise 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 realises 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 induction, 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 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 _purely_ 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 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 Cæsar." On another
occasion, Brutus and Cæsar 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 Cæsar should be
dead, and believes that he will be dead if he is stabbed; Brutus
therefore stabs him, and the stab causes Cæsar's death, as Brutus
expected it would. Every act which realises 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 in 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 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 simultaneous 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 indubitable, 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 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 call 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 realisation of
the purpose which led to the act. 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 consequences 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 feel ourselves in
the hands of fate, so that, however much we may desire to will one
thing, we may nevertheless be compelled by an outside force to will
another. We do not wish to think that, however much we may desire to act
well, heredity and surroundings may force us 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 often
predict those of other people, especially if they are elderly. Much as
we esteem the old gentleman 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: (1) Are human actions
theoretically predictable from a sufficient number of antecedents? (2)
Are human actions subject to an external compulsion? The two questions,
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.

(1) _Are human actions theoretically predictable from a sufficient
number of antecedents?_ Let us first endeavour to give precision to this
question. We may 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
determinist 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. In fact, what is found to be 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 feel,
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 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_
certainty 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 held
necessarily certain that all have causes. There are, 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 consequences 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 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 realised, 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 compulsion?_ 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 choose 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 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 civilised 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 immediately, 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 independent 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 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 foreseen 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 than 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 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 illustrations 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 generalisations, cannot make them certain. And a hasty
generalisation, such as Spencer's generalisation of evolution, is none
the less hasty because what is generalised 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 all, 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 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 philosopher'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 Metapontion was drowned for revealing it. To
us, the discovery of this fact is the chief claim of Pythagoras to
immortality, while his system has become a matter of merely historical
curiosity.[57] 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.

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

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 philosophising. So strangely perverted do men
become by unrecognised passions, that a determination in 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
agreeable 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 recognised is hard to
distinguish from vivid imaginability 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 naïve 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 behaving 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 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 superstitious
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 acquisition 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 established 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 fruitfulness he himself divined. But his few facts
sufficed to 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
consequences 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 frivolous, 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. 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 extraordinarily abstract questions underlie any one of
the big obvious problems. When everything 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 realised in the case in question. Failure to think
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 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 hypotheses which scientific philosophy aims at facilitating.
The one and only condition, I believe, which is necessary 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, 6, 39.

  Abstraction, principle of, 42, 124 ff.

  Achilles, Zeno's argument of, 173.

  Acquaintance, 25, 144.

  Activity, 224 ff.

  Allman, 161 n.

  Analysis, 185, 204, 211, 241.
    legitimacy of, 150.

  Anaximander, 3.

  Antinomies, Kant's, 155 ff.

  Aquinas, 10.

  Aristotle, 40, 160 n., 161 ff., 240.

  Arrow, Zeno's argument of, 173.

  Assertion, 52.

  Atomism, logical, 4.

  Atomists, 160.


  Belief, 58.
    primitive and derivative, 69 ff.

  Bergson, 4, 11, 13, 20 ff., 137, 138, 150, 158, 165, 174, 178, 229 ff.

  Berkeley, 63, 64, 102.

  Bolzano, 165.

  Boole, 40.

  Bradley, 6, 39, 165.

  Broad, 172 n.

  Brochard, 169 n.

  Burnet, 19 n., 160 n., 161 n., 170 n., 171 ff.


  Calderon, 95.

  Cantor, vi, vii, 155, 165, 190, 194, 199.

  Categories, 38.

  Causal laws, 109, 212 ff.
    evidence for, 216 ff.
    in psychology, 219.

  Causation, 34 ff., 79, 212 ff.
    law of, 221.
    not _a priori_, 223, 232.

  Cause, 220, 223.

  Certainty, degrees of, 67, 68, 212.

  Change,
    demands analysis, 151.

  Cinematograph, 148, 174.

  Classes, 202.
    non-existence of, 205 ff.

  Classical tradition, 3 ff., 58.

  Complexity, 145, 157 ff.

  Compulsion, 229, 233 ff.

  Congruence, 195.

  Consecutiveness, 134.

  Conservation, 105.

  Constituents of facts, 51, 145.

  Construction _v._ inference, iv.

  Contemporaries, initial, 119, 120 n.

  Continuity, 64, 129 ff., 141 ff., 155 ff.
    of change, 106, 108, 130 ff.

  Correlation of mental and physical, 233.

  Counting, 164, 181, 187 ff., 203.

  Couturat, 40 n.


  Dante, 10.

  Darwin, 4, 11, 23, 30.

  Data, 65 ff., 211.
    "hard" and "soft," 70 ff.

  Dates, 117.

  Definition, 204.

  Descartes, 5, 73, 238.

  Descriptions, 201, 214.

  Desire, 227, 235.

  Determinism, 233.

  Doubt, 237.

  Dreams, 85, 93.

  Duration, 146, 149.


  Earlier and later, 116.

  Effect, 220.

  Eleatics, 19.

  Empiricism, 37, 222.

  Enclosure, 114 ff., 120.

  Enumeration, 202.

  Euclid, 160, 164.

  Evellin, 169.

  Evolutionism, 4, 11 ff.

  Extension, 146, 149.

  External world, knowledge of, 63 ff.


  Fact, 51.
    atomic, 52.

  Finalism, 13.

  Form, logical, 42 ff., 185, 208.

  Fractions, 132, 179.

  Free will, 213, 227 ff.

  Frege, 5, 40, 199 ff.


  Galileo, 4, 59, 192, 194, 239, 240.

  Gaye, 169 n., 175, 177.

  Geometry, 5.

  Giles, 206 n.

  Greater and less, 195.


  Harvard, 4.

  Hegel, 3, 37 ff., 46, 166.

  "Here," 73, 92.

  Hereditary properties, 195.

  Hippasos, 163, 237.

  Hui Tzŭ, 206.

  Hume, 217, 221.

  Hypotheses in philosophy, 239.


  Illusions, 85.

  Incommensurables, 162 ff., 237.

  Independence, 73, 74.
    causal and logical, 74, 75.

  Indiscernibility, 141, 148.

  Indivisibles, 160.

  Induction, 34, 222.
    mathematical, 195 ff.

  Inductiveness, 190, 195 ff.

  Inference, 44, 54.

  Infinite, vi, 64, 133, 149.
    historically considered, 155 ff.
    "true," 179, 180.
    positive theory of, 185 ff.

  Infinitesimals, 135.

  Instants, 116 ff., 129, 151, 216.
    defined, 118.

  Instinct _v._ Reason, 20 ff.

  Intellect, 22 ff.

  Intelligence, how displayed by friends, 93.
    inadequacy of display, 96.

  Interpretation, 144.


  James, 4, 10, 13.

  Jourdain, 165 n.

  Jowett, 167.

  Judgment, 58.


  Kant, 3, 112, 116, 155 ff., 200.

  Knowledge about, 144.


  Language, bad, 82, 135.

  Laplace, 12.

  Laws of nature, 218 ff.

  Leibniz, 13, 40, 87, 186, 191.

  Logic, 201.
    analytic not constructive, 8.
    Aristotelian, 5.
    and fact, 53.
    inductive, 34, 222.
    mathematical, vi, 40 ff.
    mystical, 46.
    and philosophy, 8, 33 ff., 239.

  Logical constants, 208, 213.


  Mach, 123, 224.

  Macran, 39 n.

  Mathematics, 40, 57.

  Matter, 75, 101 ff.
    permanence of, 102 ff.

  Measurement, 164.

  Memory, 230, 234, 236.

  Method, deductive, 5.
    logical-analytic, v, 65, 211, 236 ff.

  Milhaud, 168 n., 169 n.

  Mill, 34, 200.

  Montaigne, 28.

  Motion, 130, 216.
    continuous, 133, 136.
    mathematical theory of, 133.
    perception of, 137 ff.
    Zeno's arguments on, 168 ff.

  Mysticism, 19, 46, 63, 95.


  Newton, 30, 146.

  Nietzsche, 10, 11.

  Noël, 169.

  Number, cardinal, 131, 186 ff.
    defined, 199 ff.
    finite, 160, 190 ff.
    inductive, 197.
    infinite, 178, 180, 188 ff., 197.
    reflexive, 190 ff.


  Occam, 107, 146.

  One and many, 167, 170.

  Order, 131.


  Parmenides, 63, 165 ff., 178.

  Past and future, 224, 234 ff.

  Peano, 40.

  Perspectives, 88 ff., 111.

  Philoponus, 171 n.

  Philosophy and ethics, 26 ff.
    and mathematics, 185 ff.
    province of, 17, 26, 185, 236.
    scientific, 11, 16, 18, 29, 236 ff.

  Physics, 101 ff., 147, 239, 242.
    descriptive, 224.
    verifiability of, 81, 110.

  Place, 86, 90.
    _at_ and _from_, 92.

  Plato, 4, 19, 27, 46, 63, 165 n., 166, 167.

  Poincaré, 123, 141.

  Points, 113 ff., 129, 158.
    definition of, vi, 115.

  Pragmatism, 11.

  Prantl, 174.

  Predictability, 229 ff.

  Premisses, 211.

  Probability, 36.

  Propositions, 52.
    atomic, 52.
    general, 55.
    molecular, 54.

  Pythagoras, 19, 160 ff., 237.


  Race-course, Zeno's argument of, 171 ff.

  Realism, new, 6.

  Reflexiveness, 190 ff.

  Relations, 45.
    asymmetrical, 47.
    Bradley's reasons against, 6.
    external, 150.
    intransitive, 48.
    multiple, 50.
    one-one, 203.
    reality of, 49.
    symmetrical, 47, 124.
    transitive, 48, 124.

  Relativity, 103, 242.

  Repetitions, 230 ff.

  Rest, 136.

  Ritter and Preller, 161 n.

  Robertson, D. S., 160 n.

  Rousseau, 20.

  Royce, 50.


  Santayana, 46.

  Scepticism, 66, 67.

  Seeing double, 86.

  Self, 73.

  Sensation, 25, 75, 123.
    and stimulus, 139.

  Sense-data, 56, 63, 67, 75, 110, 141, 143, 213.
    and physics, v, 64, 81, 97, 101 ff., 140.
    infinitely numerous? 149, 159.

  Sense-perception, 53.

  Series, 49.
    compact, 132, 142, 178.
    continuous, 131, 132.

  Sigwart, 187.

  Simplicius, 170 n.

  Simultaneity, 116.

  Space, 73, 88, 103, 112 ff., 130.
    absolute and relative, 146, 159.
    antinomies of, 155 ff.
    perception of, 68.
    of perspectives, 88 ff.
    private, 89, 90.
    of touch and sight, 78, 113.

  Spencer, 4, 12, 236.

  Spinoza, 46, 166.

  Stadium, Zeno's argument of, 134 n., 175 ff.

  Subject-predicate, 45.

  Synthesis, 157, 185.


  Tannery, Paul, 169 n.

  Teleology, 223.

  Testimony, 67, 72, 82, 87, 96, 212.

  Thales, 3.

  Thing-in-itself, 75, 84.

  Things, 89 ff., 104 ff., 213.

  Time, 103, 116 ff., 130, 155 ff., 166, 215.
    absolute or relative, 146.
    local, 103.
    private, 121.


  Uniformities, 217.

  Unity, organic, 9.

  Universal and particular, 39 n.


  Volition, 223 ff.


  Whitehead, vi, 207.

  Wittgenstein, vii, 208 n.

  Worlds, actual and ideal, 111.
    possible, 186.
    private, 88.


  Zeller, 173.

  Zeno, 129, 134, 136, 165 ff.


    PRINTED BY NEILL AND CO., LTD., EDINBURGH.



  [ Transcriber's Note:

    The following is a list of corrections made to the original.
    The first line is the original line, the second the corrected one.

    Second Impression
    Second Impression.

  more often than the impossibility of alternatives which seemed _prima
  more often than the impossibility of alternatives which seemed _primâ

  with it. We will call these the "initial contemporaries of the given
  with it. We will call these the "initial contemporaries" of the given

  the next are liable to fail when we come to infinite number. The first
  the next are liable to fail when we come to infinite numbers. The first

  of psychology or an outcome of pscyhical processes as the North Sea....
  of psychology or an outcome of psychical processes as the North Sea....

  according to the length and direction of the interval. "A quarter of a
  according to the length and direction of the interval. "A quarter of an

    Intelligence, how displayed by friends, 93
    Intelligence, how displayed by friends, 93.

    Number, cardinal, 131 186 ff.
    Number, cardinal, 131, 186 ff.

  ]





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