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Our Knowledge Of The External Wo=
rld
As A Field For Scientific Method In Philosophy
By
Bertrand Russell
Contents
LECTURE
I - CURRENT TENDENCIES
LECTURE
II - LOGIC AS THE ESSENCE OF PHILOSOPHY.
LECTURE
III - ON OUR KNOWLEDGE OF THE EXTERNAL WORLD..
LECTURE
IV - THE WORLD OF PHYSICS AND THE WORLD OF SENSE.
LECTURE
V - THE THEORY OF CONTINUITY
LECTURE
VI - THE PROBLEM OF INFINITY CONSIDERED HISTORICALLY.
LECTURE
VII - THE POSITIVE THEORY OF INFINITY.
LECTURE
VIII - ON THE NOTION OF CAUSE, WITH APPLICATIONS TO THE FREE-WILL PROBLEM
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 comp=
lete
example is to be found in the writings of Frege, has gradually, in the cour=
se
of actual research, increasingly forced itself upon me as something perfect=
ly definite,
capable of embodiment in maxims, and adequate, in all branches of philosoph=
y,
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 consider=
ed
inadmissible. Regarded merely as hypotheses and as aids to imagination, the
great systems of the past serve a very useful purpose, and are abundantly w=
orthy
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 Bos=
ton,
in March and April 1914.
The central probl=
em
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 mathemat=
ical
physics. I have been made aware of the importance of this problem by my fri=
end
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 t=
he
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 prec=
ise 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 bei=
ng
successfully carried through, a wholly new light is thrown on the time-hono=
ured
controversies of realists and idealists, and a method is obtained of solvin=
g all
that is soluble in their problem.
[2] London and New York, 1912 ("Ho=
me
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 baffle=
d,
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 practical=
ly
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 Whiteh=
ead'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 th=
eory
of infinity, no finality is claimed for the theories suggested; but I belie=
ve
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.
LECTURE I - CURRENT
TENDENCIES
Philosophy, from the earliest times, has=
made
greater claims, and achieved fewer results, than any other branch of learni=
ng.
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 hav=
e come
from other philosophers ever since Thales was contradicted by Anaximander. I
believe that the time has now arrived when this unsatisfactory state of thi=
ngs
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, be=
en
misconceived by all schools, many of its traditional problems being insolub=
le
with our means of knowledge, while other more neglected but not less import=
ant
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 attain=
ed.
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 adap=
t 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 t=
imes
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 Spenc=
er.
The third type, which may be called "logical atomism" for want of=
a
better name, has gradually crept into philosophy through the critical scrut=
iny
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 impregnat=
ed
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 for=
ce,
and it has failed to adapt itself to the temper of the age. Its advocates a=
re,
in the main, those whose extra-philosophical knowledge is literary, rather =
than
those who have felt the inspiration of science. There are, apart from reaso=
ned
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 impu=
lse
out of which the classical tradition developed was the naïve faith of the G=
reek
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 bec=
ause
they believed in the correctness of their reasoning. Thus it came to be tho=
ught
that by mere thinking the most surprising and important truths concerning t=
he
whole of reality could be established with a certainty which no contrary ob=
servations
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 Ag=
es, still
accepted more or less uncritically the Aristotelian logic. Moreover, it sti=
ll
believed, except in Great Britain, that a priori reasoning could reveal
otherwise undiscoverable secrets about the universe, and could prove realit=
y 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 re=
gard
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 takin=
g a
particular exponent as an illustration. For this purpose, let us consider f=
or a
moment the doctrines of Mr Bradley, who is probably the most distinguished
living representative of this school. Mr Bradley's Appearance and Reality i=
s a
book consisting of two parts, the first called Appearance, the second Reali=
ty.
The first part examines and condemns almost all that makes up our everyday =
world:
things and qualities, relations, space and time, change, causation, activit=
y,
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 whol=
e,
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 alternativ=
e 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 aft=
er,
father and son, and so on. But relations, according to Mr Bradley, are foun=
d on
examination to be self-contradictory and therefore impossible. He first arg=
ues
that, if there are relations, there must be qualities between which they ho=
ld. 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 s=
o,
as we saw, they have ceased to be qualities, and their relation is a nonent=
ity.
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 o=
ne
or both of its terms; or, at least, as such it seems indefensible. And, bei=
ng
something itself, if it does not itself bear a relation to the terms, in wh=
at 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 findi=
ng
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 lin=
k to
connect them with the old. The problem is to find how the relation can stan=
d to
its qualities, and this problem is insoluble."[4]
[4] Appearance and Reality, pp. 32-33.<= o:p>
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. M=
ost
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 natu=
ral
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 e=
rror
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 g=
enerally.
The function of l=
ogic
in philosophy, as I shall try to show at a later stage, is all-important; b=
ut 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 reali=
sed
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 o=
pinion,
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 revoluti=
on
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 th=
em
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 defen=
ded
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 s=
ense
would naturally object that there are people--say in China--with whom our
relations are so indirect and trivial that we cannot infer anything importa=
nt
as to them from any fact about ourselves. If there are living beings in Mar=
s or
in more distant parts of the universe, the same argument becomes even stron=
ger.
But further, perhaps the whole contents of the space and time in which we l=
ive
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 exami=
ning
is the doctrine that all reality is what is called "mental" or
"spiritual," or that, at any rate, all reality is dependent for i=
ts
existence upon what is mental. This view is often particularised into the f=
orm
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 t=
hat
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 unknow=
n, 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 sh=
ows
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 tidines=
s 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 a=
nd
Ghibelline. Hence William James's protest against what he calls the "b=
lock
universe" of the classical tradition; hence Nietzsche's worship of for=
ce;
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 philos=
ophy
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, pragmati=
sm,
Bergson, are phases in its philosophic development, and their popularity far
beyond the circles of professional philosophers shows its consonance with t=
he
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 useles=
s 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 th=
at
is important and vital to a true understanding of the universe has been
forgotten. Something of Hellenism must be combined with the new spirit befo=
re
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 mode=
l to
which all other sciences must adapt themselves. Evolutionism, as I shall tr=
y 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 mo=
re
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 a=
nd
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 achievemen=
t,
involving intermediate beings who could not with certainty be placed either
within or without the human family. The sun and planets had already been sh=
own
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 th=
eir
boundaries, and none could say where they began or where they ended.
But if human conc=
eit
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 opini=
on
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 st=
atic
to be inspiring. Not only the aspirations, but the ideal too, must change a=
nd
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 imp=
ose.
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 influenc=
e 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 gener=
al
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 gradu=
ally
approaching, is rejected by M. Bergson as not allowing enough for the absol=
ute
dominion of change. After explaining why he does not accept mechanism, he p=
roceeds:[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 thi=
ngs
and beings merely realise a programme previously arranged. But if there is
nothing unforeseen, no invention or creation in the universe, time is usele=
ss
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 fini=
te intellects
along successive things, whose successiveness is reduced to a mere appearan=
ce,
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 appearan=
ce,
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 finalis=
m 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 f=
orm
of it be put aside, it will take another. Its principle, which is essential=
ly psychological,
is very flexible. It is so extensible, and thereby so comprehensive, that o=
ne
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 finali=
sm
to a certain extent."
[5] Creative Evolution, English transla=
tion,
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. Separa=
te
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 consist=
s 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 conceiv=
ed
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, rec=
edes
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 prop=
ose
at present to enter upon a technical examination of this philosophy. At pre=
sent
I wish to make only two criticisms of it--first, that its truth does not fo=
llow
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 le=
ss
differentiated ancestry. This fact is in itself exceedingly interesting, bu=
t 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 ve=
ry
important to us as active sentient beings; but to us as philosophers they h=
ave
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 ap=
pear
to our present ethical notions to be in the nature of a progress, that give=
s 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 exis=
ts,
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 physic=
s 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 w=
hich
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 predomina=
nt
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 i=
s 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 b=
e 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 resul=
ts
which the other sciences can neither prove nor disprove.
The consideration
that philosophy, if there is such a study, must consist of propositions whi=
ch
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 l=
east
in theory, to special sciences, and are capable, at least in theory, of bei=
ng
decided by empirical evidence. Philosophers have too often, in the past,
permitted themselves to pronounce on empirical questions, and found themsel=
ves,
as a result, in disastrous conflict with well-attested facts. We must,
therefore, renounce the hope that philosophy can promise satisfaction to ou=
r 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 anal=
ysis
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 ki=
nd
of satisfaction which the other sciences offer. But it does not offer, or a=
ttempt
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 cer=
tain
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 vari=
ous
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 firs=
t 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 evolutioni=
sts.
That these conceptions stand in need of reconstruction will be admitted, an=
d 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-phil=
osophic
stage, appear to be facts, and keeping always as close to these initial dat=
a as
the requirements of consistency will permit.
Although explicit
controversy is almost always fruitless in philosophy, owing to the fact tha=
t 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 u=
nion
or the conflict of these two attitudes. Among the earliest Greek philosophe=
rs,
the Ionians were more scientific and the Sicilians more mystical.[6] But am=
ong
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 bea=
ns.
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 mathemati=
cs.
Plato, of course, embodies both the scientific and the mystical attitudes i=
n 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 sti=
ll
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 n=
ot
employ logic, which they despise: they appeal instead directly to the immed=
iate
deliverance of their insight. Now, although fully developed mysticism is ra=
re
in the West, some tincture of it colours the thoughts of many people,
particularly as regards matters on which they have strong convictions not b=
ased
on evidence. In all who seek passionately for the fugitive and difficult go=
ods,
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 wa=
y 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 bec=
omes
imperative--is that insight, untested and unsupported, is an insufficient
guarantee of truth, in spite of the fact that much of the most important tr=
uth
is first suggested by its means. It is common to speak of an opposition bet=
ween
instinct and reason; in the eighteenth century, the opposition was drawn in
favour of reason, but under the influence of Rousseau and the romantic move=
ment
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 ma=
inly
illusory. Instinct, intuition, or insight is what first leads to the beliefs
which subsequent reason confirms or confutes; but the confirmation, where i=
t 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 fi=
rst
arrives at what is new.
Where instinct and
reason do sometimes conflict is in regard to single beliefs, held instincti=
vely,
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 insti=
nct
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 o=
ften
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 t=
his
there is no opposition to instinct as a whole, but only to blind reliance u=
pon
some one interesting aspect of instinct to the exclusion of other more comm=
onplace
but not less trustworthy aspects. It is such onesidedness, not instinct its=
elf,
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 t=
hat
we move round the object; the second that we enter into it. The first depen=
ds
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 flow=
ing
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.<= o:p>
This procedure, s=
ince
it takes sides in a conflict of instinctive beliefs, stands in need of just=
ification
by proving the greater trustworthiness of the beliefs on one side than of t=
hose
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, a=
nd
by pointing out characteristics of the world which, though intuition can
apprehend them, are baffling to intellect as he interprets it.
Of Bergson's theo=
ry
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 o=
nly
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 th=
is
merely inferred history is presumably untrue. If, on the other hand, we agr=
ee
with M. Bergson in thinking that evolution took place as Darwin believed, t=
hen 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 knowl=
edge
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 socie=
ties
men are not put to death for mathematical incompetence. All the most striki=
ng
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 t=
hey
give truth and become harmful when they give falsehood. Intellect, in civil=
ised
man, like artistic capacity, has occasionally been developed beyond the poi=
nt
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 runnin=
g wild
in the woods, dyeing themselves with woad and living on hips and haws.
Let us next exami=
ne
whether intuition possesses any such infallibility as Bergson claims for it.
The best instance of it, according to him, is our acquaintance with ourselv=
es;
yet self-knowledge is proverbially rare and difficult. Most men, for exampl=
e,
have in their nature meannesses, vanities, and envies of which they are qui=
te
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 intellec=
t, 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 gradu=
ally
proves, as a rule, that the supposed insight was illusory, and that the slo=
wer,
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 ha=
s 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 duckling=
s 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, l=
ike
all instinct, is admirable in those customary surroundings which have mould=
ed
the habits of the animal in question, but totally incompetent as soon as th=
e 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 civil=
ised
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 o=
ur kinship
with remote generations of animal and semi-human ancestors, that show intui=
tion
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 refine=
d,
highly civilised pursuit, demanding, for its success, a certain liberation =
from
the life of instinct, and even, at times, a certain aloofness from all mund=
ane
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 philosoph=
y,
and the habits of thought demanded for their apprehension, are strange,
unusual, and remote, it is here, more almost than anywhere else, that intel=
lect
proves superior to intuition, and that quick unanalysed convictions are lea=
st
deserving of uncritical acceptance.
Before embarking =
upon
the somewhat difficult and abstract discussions which lie before us, it wil=
l be
well to take a survey of the hopes we may retain and the hopes we must aban=
don.
The hope of satisfaction to our more human desires--the hope of demonstrati=
ng
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 partic=
ular
characteristics of the particular things that exist in these worlds: it is =
not
a sufficiently abstract difference to come within the province of philosoph=
y.
Love and hate, for example, are ethical opposites, but to philosophy they a=
re
closely analogous attitudes towards objects. The general form and structure=
of
those attitudes towards objects which constitute mental phenomena is a prob=
lem
for philosophy; but the difference between love and hate is not a differenc=
e 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 o=
r be
expected in the special results which are sought.
If this view seem=
s 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 whi=
ch
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 disinte=
rested
study of astronomy began, many who had found astrology absorbingly interest=
ing
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, w=
ith
its ethical attributes: he is merely concerned to find out facts, not to
consider whether they are good or bad. In psychology, the scientific attitu=
de
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 ethica=
lly
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 achieve=
d.
Men have remembered their wishes, and have judged philosophies in relation =
to
their wishes. Driven from the particular sciences, the belief that the noti=
ons
of good and evil must afford a key to the understanding of the world has so=
ught
a refuge in philosophy. But even from this last refuge, if philosophy is no=
t 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 direct=
ly;
and it would seem that the same is true of the good. In thought, at any rat=
e,
those who forget good and evil and seek only to know the facts are more lik=
ely
to achieve good than those who view the world through the distorting medium=
of
their own desires.
The immense exten=
sion
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 inadequ=
ate
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 makesh=
ift.
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 increa=
se
in the future beyond all easily assignable limits. Thus alongside of despai=
r as
regards ultimate theory there is an immense optimism as regards practice: w=
hat
man can do seems almost boundless. The old fixed limits of human power, suc=
h 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 larg=
e,
the modern spirit should, I think, be accepted as wholly an advance. But bo=
th
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, a=
nd
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, t=
herefore,
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 t=
he
practical problems of life. To those who wish to understand much of what ha=
s in
the past been most difficult and obscure in the constitution of the univers=
e,
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 investigatio=
n always
does--a sense of power and a hope of progress more reliable and better grou=
nded
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 mi=
nds.
LECTURE II - LOGIC AS THE
ESSENCE OF PHILOSOPHY
The topics we discussed in our first lec=
ture,
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 us=
ing
the word, logical. But as the word "logic" is never used in the s=
ame
sense by two different philosophers, some explanation of what I mean by the
word is indispensable at the outset.
Logic, in the Mid=
dle
Ages, and down to the present day in teaching, meant no more than a scholas=
tic
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 w=
hich
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 ha=
ve
widened the scope of logic.
The first extensi=
on
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 t=
he
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 w=
ill
remain merely as one of the principles according to which deductions are
effected. Thus the ultimate result of the introduction of the inductive met=
hod
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 t=
he
scope and validity of induction is of great difficulty, and of great import=
ance
to our knowledge. Take such a question as, "Will the sun rise to-morro=
w?"
Our first instinctive feeling is that we have abundant reason for saying th=
at
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 princi=
ple of
inference by which we pass from past sunrises to future ones? The answer gi=
ven
by Mill is that the inference depends upon the law of causation. Let us sup=
pose
this to be true; then what is the reason for believing in the law of causat=
ion?
There are broadly three possible answers: (1) that it is itself known a pri=
ori;
(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 causatio=
n 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 theo=
ry
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 can=
not
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 say=
s 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, "c=
onsists
in ascribing the nature of general truths to all propositions which are tru=
e in
every instance that we happen to know of."[8] As regards its fallibili=
ty,
he asserts that "the precariousness of the method of simple enumeratio=
n 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 t=
he 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 clas=
s 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 ta=
ke
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 val=
id,
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 ever=
y instance
we have been able to test; therefore it probably holds in untested instance=
s.
There are terrible difficulties in the notion of probability, but we may ig=
nore
them at present. We thus have what at least may be a logical principle, sin=
ce
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 s=
ay,
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 probab=
le
does not always happen, for an event may be probable on the data and yet no=
t 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 princip=
le
as this is required if the method of simple enumeration is to be valid.
[10] Or rather a propositional function=
.
But this brings u=
s to
our other question, namely, how is our principle known to be true? Obviousl=
y,
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 su=
ch
data. Hence, if it is known, it is not known by experience, but independent=
ly
of experience. I do not say that any such principle is known: I only say th=
at
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 induc=
tion
will be discussed again in Lecture
VIII.
A similar conclus=
ion
can be proved by similar arguments concerning any other logical principle. =
Thus
logical knowledge is not derivable from experience alone, and the empiricis=
t'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 advoc=
ate.
In their writings, logic is practically identical with metaphysics. In broad
outline, the way this came about is as follows. Hegel believed that, by mea=
ns
of a priori reasoning, it could be shown that the world must have various i=
mportant
and interesting characteristics, since any world without these characterist=
ics
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 univer=
se.
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 applicati=
on
of logic to the actual world. Logic itself would be concerned rather with s=
uch
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 th=
e traditional
logic, with all its faults, is uncritically and unconsciously assumed
throughout his reasoning. It is not in the direction advocated by him, it s=
eems
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 whi=
ch
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 subseque=
ntly
criticises, is exemplified by the general conception of "categories&qu=
ot;
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 wo=
uld
not ascribe a predicate to either. Thus Hegel's doctrine, that philosophica=
l 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. T=
his
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 tru=
th.
This is the most important respect in which Hegel uncritically assumes the
traditional logic. Other less important respects--though important enough t=
o be
the source of such essentially Hegelian conceptions as the "concrete u=
niversal"
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. Macra=
n,
Hegel's Doctrine of Formal Logic,
Oxford, 1912. Hegel's argument in this portion of his "Logic"
There is quite an=
other
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 k=
ind
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 bra=
nch
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 amazi=
ng
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 po=
ints
in the traditional doctrine of the syllogism. We now know that on these poi=
nts
the traditional doctrine is wrong, but respect for Aristotle prevented Leib=
niz
from realising that this was possible.[13]
[13] Cf. Couturat, La Logique de Leibni=
z, 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 meth=
ods
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 arriv=
ed at
their logical results by an analysis of mathematics. Traditional logic rega=
rded
the two propositions, "Socrates is mortal" and "All men are
mortal," as being of the same form;[14] Peano and Frege showed that th=
ey
are utterly different in form. The philosophical importance of logic may be
illustrated by the fact that this confusion--which is still committed by mo=
st
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 impossib=
le
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 logi=
c,
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 fou=
nd
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 en=
able
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 inspi=
red
by mathematical logic, and could never have been imagined without it.
In both these cas=
es,
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 o=
ne
which clears away incredible accumulations of metaphysical lumber, was dire=
ctly
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 gro=
up
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 member=
ship
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 no=
t be
assumed to exist. In this and other ways, the indirect uses of even the lat=
er
parts of mathematical logic are very great; but it is now time to turn our
attention to its philosophical foundations.
In every proposit=
ion
and in every inference there is, besides the particular subject-matter
concerned, a certain form, a way in which the constituents of the propositi=
on
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 repl=
ace
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 constitue=
nts
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 clo=
se
relation to existing things that drinking has. It is something altogether m=
ore
abstract and remote. We might understand all the separate words of a senten=
ce
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 knowledg=
e of
the constituents. If I say, "Rorarius drank the hemlock," those a=
mong
you who have never heard of Rorarius (supposing there are any) will underst=
and
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 a=
ll
understanding of discourse. It is the business of philosophical logic to
extract this knowledge from its concrete integuments, and to render it expl=
icit
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 a=
re
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, a=
nd
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 ki=
nd
of generality. If they seem to depend upon the subject-matter otherwise tha=
n 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 inferen=
ces
concerning particular cases: we deal throughout with completely general and
purely formal implications, leaving it to other sciences to discover when t=
he
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 proposition=
s),
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 f=
orm,
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 sa=
ying
"that thing is smaller than this," where grammatically the subjec=
t is
changed. Thus propositions stating that two things have a certain relation =
have
a different form from subject-predicate propositions, and the failure to pe=
rceive
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; b=
ut
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 wh=
o 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 pro=
ved
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 doctr=
ines
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 use=
ful
word from Mr Santayana--"malicious" in regard to the world of sci=
ence
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 t=
he daily
world to which it is to be applied. Such an attitude naturally does not ten=
d 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 mu=
st
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 un=
able
to admit the reality of relations: all relations, it maintains, must be red=
uced
to properties of the apparently related terms. There are many ways of refut=
ing
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, w=
hen
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, t=
hen
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 cal=
led
asymmetrical when, if it holds between A and B, it never holds between B an=
d 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 in=
to
symmetrical, asymmetrical, and merely non-symmetrical relations is the firs=
t 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 sai=
d 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 sai=
d 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 kind=
s of
dissimilarity are non-transitive.
A relation is sai=
d to
be intransitive when, if A has the relation to B, and B to C, A never has i=
t 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 doctri=
ne.
A symmetrical relation which is transitive, such as equality, can be regard=
ed
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 bec=
omes
obviously impossible. When, for example, two things are merely known to be
unequal, without our knowing which is greater, we may say that the inequali=
ty
results from their having different magnitudes, because inequality is a
symmetrical relation; but to say that when one thing is greater than anothe=
r,
and not merely unequal to it, that means that they have different magnitude=
s,
is formally incapable of explaining the facts. For if the other thing had b=
een
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 everyth=
ing
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 frank=
ly
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 poss=
ibility
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 w=
hen
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 h=
ave
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 recei=
ved
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 betw=
een
three people. Professor Royce mentions the relation "giving": whe=
n 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, w=
hich
yellow, which was earlier than which, which was between which two others, a=
nd
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 bet=
ween
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 t=
wo
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 a=
nd 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 t=
hree
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 mus=
t 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 differe=
nt
numbers of things have different forms.
Given any fact, t=
here
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 someth=
ing
which involves thought, and may be either true or false. An assertion may b=
e 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 expres=
ses
what we have called a fact, i.e. which, when asserted, asserts that a certa=
in
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 propositio=
ns
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 atom=
ic
fact may sometimes be capable of being inferred from another, though this s=
eems
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 c=
ome
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 w=
hat
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 atom=
ic
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 be=
liefs
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&q=
uot;
propositions are such as contain conjunctions--if, or, and, unless, etc.--a=
nd
such words are the marks of a molecular proposition. Consider such an asser=
tion
as, "If it rains, I shall bring my umbrella." This assertion is j=
ust
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 ca=
se
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 d=
oes
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 p=
ractical
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" occur=
s,
such as "some men are philosophers" or "some philosophers are
not wise." These are the denials of general propositions, namely (in t=
he
above instances), of "all men are non-philosophers" and "all=
philosophers
are wise." We will call propositions containing the word "some&qu=
ot;
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 i=
nfer
all other truths by logic if we knew all atomic facts and also knew that th=
ere
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 th=
at
"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,&qu=
ot;
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 kn=
ow
that all men are mortal, unless we knew that those were all the men there a=
re,
which is a general proposition. If we knew every other existing thing
throughout the universe, and knew that each separate thing was not an immor=
tal
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 collect=
ion
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 conclus=
ion,
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 up=
on
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 t=
hat
this should be the case, but it does not appear to be so in fact, and indee=
d no
one would dream of maintaining such a view except a theorist at the last ex=
tremity.
We must therefore admit that there is general knowledge not derived from se=
nse,
and that some of this knowledge is not obtained by inference but is primiti=
ve.
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 Soc=
rates
is mortal," because Socrates and man and mortal are empirical terms, o=
nly
to be understood through particular experience. The corresponding propositi=
on
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 do=
es
not mention any particular thing, or even any particular quality or relatio=
n,
it is wholly independent of the accidental facts of the existent world, and=
can
be known, theoretically, without any experience of particular things or the=
ir
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 mer=
ges
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 t=
ruly
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 alrea=
dy
seen how the supposed universality of the subject-predicate form made it
impossible to give a right analysis of serial order, and therefore made spa=
ce
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 judg=
ment
consisted in apprehension of a fact, and that the apprehension was a relati=
on
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 objec=
tive
fact "Charles I.'s death in his bed" to which I can have a relati=
on
of apprehension. Charles I. and death and his bed are objective, but they a=
re
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 opin=
ion,
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 t=
he
logic practised by the classical tradition. In that logic, hypotheses which
seem primâ facie possible are professedly proved impossible, and it is decr=
eed
in advance that reality must have a certain special character. In modern lo=
gic,
on the contrary, while the primâ facie hypotheses as a rule remain admissib=
le,
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 roa=
ds,
but one of the oldest and most travelled is the road which leads through do=
ubt
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 bewilde=
ring
variety of motives. The mystic condemns it on the ground of immediate knowl=
edge
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 lo=
ng
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 mean=
s 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 m=
ay
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 a=
ny
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 m=
ethod
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 v=
ery
powerful. I think it must be admitted as probable that the immediate object=
s of
sense depend for their existence upon physiological conditions in ourselves,
and that, for example, the coloured surfaces which we see cease to exist wh=
en
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 knowle=
dge
of the external world. This line of argument will be developed in the prese=
nt lecture.
The discrepancy
between the world of physics and the world of sense, which we shall conside=
r in
our fourth lecture, will be found to be more apparent than real, and it wil=
l 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 log=
ic
of the text-books and also from the logic of idealism. Our second lecture h=
as
given a short account of modern logic and of its points of divergence from =
the
various traditional kinds of logic.
In our last lectu=
re,
after a discussion of causality and free will, we shall try to reach a gene=
ral
account of the logical-analytic method of scientific philosophy, and a
tentative estimate of the hopes of philosophical progress which it allows u=
s 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. Wha=
t 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 sou=
ght.
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, compl=
ex,
inexact, as common knowledge always is, but yet somehow commanding our asse=
nt
as on the whole and in some interpretation pretty certainly true. In the ca=
se
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 m=
ay
be errors of detail in this knowledge, but we believe them to be discoverab=
le
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 analy=
sis.
It may be said--a=
nd
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 se=
nse
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 who=
le
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 s=
ay,
its criticism of details will only be based upon their relation to other de=
tails,
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 a=
ny
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 questi=
on.
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 Napole=
on, 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 t=
hat
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 entanglemen=
t 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 sen=
se constantly
arise without any process of logical inference, merely by association of id=
eas
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 fac=
t 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 ph=
ysical
mark of emotion we actually saw. In such a case, the knowledge is derivativ=
e 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 arriv=
ed
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 u=
pon
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 t=
ruth
tends to diminish the more we think about them. We naturally believe, for e=
xample,
that tables and chairs, trees and mountains, are still there when we turn o=
ur
backs upon them. I do not wish for a moment to maintain that this is certai=
nly
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 th=
at
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 th=
e 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 sens=
e:
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 primit=
ive.
We are thus led t=
o a
somewhat vague distinction between what we may call "hard" data a=
nd
"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 influ=
ence
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 t=
he
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 mean=
s,
the more luminously certain do they become. Verbal doubt concerning even th=
ese
is possible, but verbal doubt may occur when what is nominally being doubte=
d is
not really in our thoughts, and only words are actually present to our mind=
s.
Real doubt, in these two cases, would, I think, be pathological. At any rat=
e,
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 univer=
sal
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 psychologicall=
y derivative
but logically primitive beliefs, we shall find that most, if not all, are t=
o 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 o=
n a
level with the facts of sense or the laws of logic. The kind of respect whi=
ch
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. Als=
o,
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 h=
ypothesis
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 discover=
ing
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 sle=
nder
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 purpose=
s,
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 l=
ed
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 bec=
ome
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 th=
e 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 arrive=
d 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 extern= al world, and to remove various misunderstandings which have obscured the mean= ing 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.<= o:p>
When we speak of =
the
"external" world in this discussion, we must not mean "spati=
ally
external," unless "space" is interpreted in a peculiar and
recondite manner. The immediate objects of sight, the coloured surfaces whi=
ch
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&=
quot;;
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 c=
an
certainly be estimated approximately by means of the data of sense alone. T=
he
immediately given world is spatial, and is further not wholly contained wit=
hin
our own bodies. Thus our knowledge of what is external in this sense is not
open to doubt.
Another form in w=
hich
the question is often put is: "Can we know of the existence of any rea=
lity
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) t=
he
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. T=
he 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 thin=
gs
depend upon our lives for their existence. And in this form, the definition=
of
Self introduces the word "depend," which raises the same question=
s 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 logica=
lly
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 t=
he 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?" reduc=
es
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 ad=
mit
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 witho=
ut
the other. Now it is fairly obvious that, whatever legitimate meaning we gi=
ve
to the Self, our thoughts and feelings are causally dependent upon ourselve=
s,
i.e. do not occur when there is no Self for them to belong to. But in the c=
ase
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 obj=
ects
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 wo=
rd
"independent" no longer occurs, is the form in which we stated the
problem a minute ago.
Our question in t=
he
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 object=
s 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 awa=
re
in sensation. When I speak of the sensible object, it must be understood th=
at I
do not mean such a thing as a table, which is both visible and tangible, ca=
n 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 sens=
ible
object, and our awareness of it I call a sensation. Now our sense of passiv=
ity,
if it really afforded any argument, would only tend to show that the sensat=
ion
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 sensi=
ble
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 somet=
hing
which can persist independently of our consciousness makes itself known in
sensation, with the fact that our sensations often change in ways which see=
m 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 seei=
ng
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 extr=
aordinarily
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 fr=
om
one place presents a different appearance from that which it presents from
another place. This is the language of common sense, but this language alre=
ady
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 elem=
ent
of hypothesis. We find that as we walk round the table, we perceive a serie=
s of
gradually changing visible objects. But in speaking of "walking round =
the
table," we have still retained the hypothesis that there is a single t=
able
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 strik=
ing
patch of colour is not suddenly replaced by something wholly different, but=
is
replaced by an insensible gradation of slightly different colours with slig=
htly
different shapes. This is what we really know by experience, when we have f=
reed
our minds from the assumption of permanent "things" with changing
appearances. What is really known is a correlation of muscular and other bo=
dily
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 sens=
e,
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 beyo=
nd
sensible objects. Anything intervening between ourselves and what we see mu=
st
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 s=
ee 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 di=
scovery
that the intervening medium affects the appearances of things cannot be mad=
e by
means of the sense of sight alone.
Let us take the c=
ase
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 glas=
s,
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 term=
s 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 sa=
y,
"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 i=
s 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 kin=
d as
objects formerly given in sense.
It may be questio=
ned,
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 sensib=
le
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 ha=
ve
effects, but this is a mere prejudice, due to a wrong conception of causali=
ty.
We cannot, therefore, dismiss our present hypothesis on the ground of a pri=
ori
impossibility, but must examine further whether it can really account for t=
he
facts.
It may be said th=
at
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 object=
s?
And more generally, what are we to make of the hypothetical sensations of t=
ouch
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 t=
he
more general question first. Experience has taught us that where we see cer=
tain
kinds of coloured surfaces we can, by touch, obtain certain expected sensat=
ions
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 wi=
ll
supply an interpretation of common-sense beliefs without assuming anything
beyond the existence of sensible objects at the times when they are sensibl=
e.
By experience of the correlation of touch and sight sensations, we become a=
ble
to associate a certain place in touch-space with a certain corresponding pl=
ace
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 h=
as
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 objec=
ts,
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-d=
ata
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 be=
en
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 oft=
en
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 l=
ook
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 moo=
n 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 con=
nection
is obtained by imagining a "real" moon which goes on whether I lo=
ok
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 th=
at,
at our present level of doubt, we are not at liberty to accept testimony. W=
hen
we hear certain noises, which are those we should utter if we wished to exp=
ress
a certain thought, we assume that that thought, or one very like it, has be=
en
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 f=
eelings
inside it continue when we are not looking at it. When we see our friend dr=
op 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 tha=
t 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 on=
ly
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 i=
s 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 i=
s:
"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 independen=
t of
ourselves?" because of the extreme difficulty of defining "self&q=
uot;
and "independent" precisely. The felt passivity of sensation is
irrelevant, since, even if it proved anything, it could only prove that sen=
sations
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 regard=
s as
the appearance of one object changes with what common sense regards as chan=
ges
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 cal=
l 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 out=
come
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 th=
at
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 statem=
ent
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 rea=
d,
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 gi=
ven
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 objec=
t is
something quite unlike that object, something which, together with us and o=
ur
sense-organs, causes our sensations, but is never itself given in sensation=
.
The thing-in-itse=
lf,
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 der=
ives
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 exi=
sts
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 th=
e 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 rou=
nd
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 theo=
ry
upon which physical science and physiology are built, and it must, therefor=
e,
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 wit=
h a
"real" visit to America. Objects of sense are called "real&q=
uot;
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 obje=
cts
of waking life. And conversely, the sensible objects of waking life must no=
t 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 equ=
al
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 sha=
ll
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 ta=
ble.
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 warran=
ted
in saying is that, in this case, the manner of correlation of touch and sig=
ht
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 s=
ame
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 o=
ne
possible (not necessary) explanation of the facts. It may perhaps then be
possible to pare away what is superfluous in our hypothesis, leaving a resi=
due
which may be regarded as the abstract answer to our problem.
Let us imagine th=
at
each mind looks out upon the world, as in Leibniz's monadology, from a poin=
t 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 se=
nse.
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 testi=
mony,
but as we are only constructing a possible theory, that is a legitimate
assumption.) The three-dimensional world seen by one mind therefore contain=
s no
place in common with that seen by another, for places can only be constitut=
ed
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 a=
s 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 perspec=
tives.
Two men are somet=
imes
found to perceive very similar perspectives, so similar that they can use t=
he
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 gre=
at
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 perspect=
ives
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 simil=
ar
objects. Given an object in one perspective, form the system of all the obj=
ects
correlated with it in all the perspectives; that system may be identified w=
ith
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 th=
eory
of relativity; but we may ignore these at present.) All the aspects of a th=
ing
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 bein=
g 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 observ=
ed
that, while each perspective contains its own space, there is only one spac=
e in
which the perspectives themselves are the elements. There are as many priva=
te
spaces as there are perspectives; there are therefore at least as many as t=
here
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 sp=
ace.
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 cou=
nt
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 thi=
s 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 th=
at
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 wi=
ll
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 spac=
e,
and that experience shows that the same spatial order of perspectives would
have resulted.
In order to expla=
in
the correlation of private spaces with perspective space, we have first to
explain what is meant by "the place (in perspective space) where a thi=
ng
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 p=
enny
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 li=
ne
of a certain thickness. These two lines will meet in a certain place in per=
spective
space, i.e. in a certain perspective, which may be defined as "the pla=
ce
(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 pen=
ny
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 or=
der
of perspectives is found empirically to be independent of the particular
"things" chosen for defining the order. We can, for example, remo=
ve
our penny and prolong each of our two straight lines up to their intersecti=
on
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 aspec=
ts
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 loo=
ks
circular and the other straight. This will be, by definition, the place whe=
re
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 remo=
ve
the penny without being disturbed by any simultaneous changes in the positi=
ons
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 sp=
ace.
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 o=
ur
private world. Thus we can now understand what is meant by speaking of a th=
ing
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 w=
hat
is meant by saying that our private world is inside our head; for our priva=
te
world is a place in perspective space, and may be part of the place where o=
ur
head is.
It will be observ=
ed
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 t=
hing
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 whi=
ch the
thing has the given aspect. The physicist naturally classifies aspects in t=
he
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 distingu=
ish
the two places as that at which, and that from which, the aspect appears. T=
he
"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 endeav=
our
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 w=
orld
we have constructed can, with a certain amount of trouble, be used to inter=
pret
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 a=
nything
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 t=
he
existence of other minds.
It must be conced=
ed
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, re=
fuse
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 n=
o 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 possib=
le
that what we call waking life may be only an unusually persistent and recur=
rent
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, monthl=
y,
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 bel=
ief
is natural to us. It is, however, a psychologically derivative belief, sinc=
e it
results from observation of people's bodies; and along with other such beli=
efs,
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 argum=
ent
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 natura=
l 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 t=
he
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 scen=
e,
with our inference, may occur in a dream, in which case the inference is ge=
nerally
considered to be mistaken. Is there anything to make the argument from anal=
ogy
more cogent when we are (as we think) awake?
The analogy in wa=
king
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 old=
er
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 spi=
rits
of the dead visit us while we sleep; but the modern mind, as a rule, refuse=
s 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 fro=
m a
sleep which has filled all his mundane life: the whole world of sense becom=
es
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 th=
at
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 nothi=
ng
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 wo=
rlds
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 mea=
gre
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 t=
hat
the testimony of others is to be admitted, and therefore (at least by
implication) that others have minds. Their difficulties have arisen after t=
his 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 doub=
tful
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 initi=
al
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 diminut=
ion
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 o=
bjects
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 f=
rom
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 a=
re
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 tha=
t I
can hope to do is to make the problem felt, and to indicate the kind of met=
hods
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 worl=
d 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 start=
ed
from the common-sense belief in fairly permanent and fairly rigid
bodies--tables and chairs, stones, mountains, the earth and moon and sun. T=
his common-sense
belief, it should be noticed, is a piece of audacious metaphysical theorisi=
ng;
objects are not continually present to sensation, and it may be doubted whe=
ther
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 there=
fore
hitherto been ignored by physicists. We have thus here a first departure fr=
om
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 prehistor=
ic
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 cl=
eft
by earthquakes and eruptions. Then there are other things, which seem mater=
ial,
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 rive=
rs
and seas, though fairly permanent, are not in any degree rigid. Breath, smo=
ke, 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 b=
ut
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 thi=
ng
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 id=
eal
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 vi=
ew
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 eve=
ry
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-embrac=
ing
space in which the motion takes place, and until lately we might have assum=
ed
one all-embracing time also. But the principle of relativity has given prom=
inence
to the conception of "local time," and has somewhat diminished me=
n'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 diffe=
rent
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 i=
n 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 d=
ata
when we see them, and are not immediately given as existing at other moment=
s.
So far from one all-embracing space being given, there are several spaces f=
or
each person, according to the different senses which give relations that ma=
y 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, b=
ut
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 t=
he
changes of the sensible world there must be something permanent and unchang=
ing.
This belief was, no doubt, fostered and nourished by its practical successe=
s,
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 we=
re 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 be=
en
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 supp=
osition
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 ca=
ll
water. We can give laws according to which the one appearance will be succe=
eded
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 worl=
d of
sense with the world of physics, is the task of reconstructing the concepti=
on
of matter without the a priori beliefs which historically gave rise to it. =
In
spite of the revolutionary results of modern physics, the empirical success=
es
of the conception of matter show that there must be some legitimate concept=
ion which
fulfils roughly the same functions. The time has hardly come when we can st=
ate
precisely what this legitimate conception is, but we can see in a general w=
ay
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 contin=
uous
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 w=
hat
it is at another. But what do we really know about it? We know that under s=
uitable
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 se=
ries
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 cer=
tain
thing will merely mean that it is one of those which, taken serially, are t=
he
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 extrusi=
on
of permanent things affords an example of the maxim which inspires all
scientific philosophising, namely "Occam's razor": Entities are n=
ot
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 changi=
ng
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 briefl=
y 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, s=
ome
conflict between what common sense regards as one thing, and what physics
regards an unchanging collection of particles. To common sense, a human bod=
y 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 se=
lect
certain data from the chaos, and call them all appearances of the same thin=
g?
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 hesitatio=
n 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 l=
ed
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 ex=
act
similarity.
Another insuffici=
ent
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 continuo=
us
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 cha=
nge
is necessary and sufficient to constitute one thing. But in fact it is neit=
her.
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 a=
nd
later appearances to belong to the same thing; on the contrary, it is becau=
se
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 m=
ark
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 prov=
es
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 l=
ack
of observation they may not always appear continuous. In this hypothetical
sense, continuity may be allowed to be a necessary condition if two appeara=
nces
are to be classed as appearances of the same thing. But it is not a suffici=
ent condition,
as appears from the instance of the drops in the sea. Thus something more m=
ust
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. T=
his
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 t=
he
same time provided the connection is not logically demonstrable. In this ve=
ry
general sense, the laws of dynamics are causal laws, and so are the laws
correlating the simultaneous appearances of one "thing" to differ=
ent
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 succe=
ss
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 calculab=
le
from a sufficient collection of data all belonging to a given period of tim=
e.
Now physics has found it empirically possible to collect sense-data into
series, each series being regarded as belonging to one "thing," a=
nd
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 unambigu=
ous
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 p=
hysics.
It would be very difficult to prove that this is the case, but for our pres=
ent
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: Thin=
gs
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 o=
bjected
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 exis=
ts
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 su=
ch
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 w=
hen,
in fact, they are not appearing to anyone; (γ) things which never appe=
ar
at all. All these are introduced to simplify the statement of the causal la=
ws,
but none of them form an integral part of what is known to be true in physi=
cs.
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 bei=
ng
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 of=
ten 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 wor=
ld
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 calculate=
d,
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 id=
eal
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 function=
s 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 log=
ical
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 i=
t.
It is open to us to believe that the ideal elements exist, and there can be=
no
reason for disbelieving this; but unless in virtue of some a priori law we
cannot know it, for empirical knowledge is confined to what we actually
observe.
(2) The three main
conceptions of physics are space, time, and matter. Some of the problems ra=
ised
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 n=
ever
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 psychologi=
cal
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&q=
uot;
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 s=
how
the problems, without taking sides on any psychological issue still in deba=
te.
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-sensatio=
ns
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 to=
uch
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 ar=
e no
infinitesimal sense-data: any surface we can see, for example, must be of s=
ome
finite extent. But what at first appears as one undivided whole is often fo=
und,
under the influence of attention, to split up into parts contained within t=
he whole.
Thus one spatial object may be contained within another, and entirely enclo=
sed
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 wo=
uld
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 o=
ne
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 tri=
vial
exceptions, we must add that there are to be instances of enclosure, i.e. t=
here
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 th=
ere
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 me=
mber
of our second series. In this case, our first enclosure-series may be calle=
d a
"punctual enclosure-series." Then a "point" is all the
objects which enclose members of a given punctual enclosure-series. In orde=
r to
ensure infinite divisibility, we require one further property to be added t=
o 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 pro=
duce
sensations which are not merely and strictly instantaneous, and therefore t=
he objects
of sense of which we are immediately conscious are not strictly instantaneo=
us.
Instants, therefore, are not among the data of experience, and, if legitima=
te,
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 experie=
nce
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-ord=
er
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 revolv=
er
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 deliber=
ate 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 w=
ould
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 g=
iven
in immediate experience that the words one and two come earlier than the fl=
ash.
These time-relations hold between events which are not strictly instantaneo=
us.
Thus one event may begin sooner than another, and therefore be before it, b=
ut
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 oth=
er
when we are concerned with events which last for a finite time, however sho=
rt;
they only become inconsistent when we are dealing with something instantane=
ous.
It is to be obser=
ved
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 ev=
ent
occurring at that time. There is therefore no reason in experience to suppo=
se
that there are times as opposed to events: the events, ordered by the relat=
ions
of simultaneity and succession, are all that experience provides. Hence, un=
less
we are to introduce superfluous metaphysical entities, we must, in defining=
what
mathematical physics can regard as an instant, proceed by means of some con=
struction
which assumes nothing beyond events and their temporal relations.
If we wish to ass=
ign
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 mus=
t 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 a=
fter
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, bu=
t B ends
before A ends. Then an event which is simultaneous with both A and B must e=
xist
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 end=
ed.
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 t=
his
way, by taking more and more events, a new event which is dated as simultan=
eous
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 gro=
up
of events of which any two overlap, so that there is some time, however sho=
rt,
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 constructe=
d 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 h=
as
the properties we expect of an instant.
What are the
properties we expect of instants? First, they must form a series: of any tw=
o,
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 t=
he
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 so=
me
instant at which the other is. Thirdly, if we assume that there is always s=
ome
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, ha=
ve
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 ano=
ther
if the group which is the one instant contains an event which is earlier th=
an,
but not simultaneous with, some event in the group which is the other insta=
nt.
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 even=
ts
which are not simultaneous, there must be one which wholly precedes the oth=
er,
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 de=
duce
that the instants as we have defined them form a series.
We have next to s=
how
that every event is "at" at least one instant, i.e. that, given a=
ny
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 seri=
es
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 f=
orm a
series, we assume:
(a) No event wholly precedes itself=
. (An
"event" is defined as whatever is simultaneous with somet=
hing
or other.)
(b) If one event wholly precedes an=
other,
and the other wholly precedes=
a
third, then the first wholly precedes the third.
(c) If one event wholly precedes an=
other,
it is not simultaneous with i=
t.
(d) Of two events which are not
simultaneous, one must wholly precede the other.
II. In order to secure that the initi=
al
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 contempor=
ary of
the given event.
III. In order to secure that the seri=
es of
instants shall be compact, we a=
ssume:
(f) If one event wholly precedes an=
other,
there is an event wholly afte=
r 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 co=
mmon
with the other event; i.e. it is
impossible for one event to cease just before another begins. I do not know whether this shou=
ld be
regarded as inadmissible. For a
mathematico-logical treatment of the above topics, cf. N. Wilner, "A Contribution to the The=
ory of
Relative Position," Proc. Ca=
mb.
Phil. Soc., xvii. 5, pp. 441-449.
Thus our definiti=
on
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 e=
ncloses
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 a=
ll
of them, then there is an event enclosing all that they all enclose, and it=
self
enclosed by all of them; (4) that there is at least one event. To ensure in=
finite
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 whi=
ch
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 comm=
on
sense would regard as appearances of the same "thing." When two
appearances in different worlds are so correlated as to belong to one momen=
tary
"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 necessar=
ily
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 techni=
cal
problems; but from the point of view of philosophical theory, there is no v=
ery
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 ki=
nd
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 ca=
n be
called respectively particles, points, and instants. If such constructions =
are
possible, then mathematical physics is applicable to the real world, in spi=
te
of the fact that its particles, points, and instants are not to be found am=
ong
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 stu=
dies
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 l=
ight
the chaotic nature of the crude materials supplied by unmanipulated sensati=
on,
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 wh=
ich
physics shows them to possess. Philosophers, it is to be hoped, will come t=
o recognise
that they cannot achieve any solid success in such problems without some sl=
ight
knowledge of logic, mathematics, and physics; meanwhile, for want of studen=
ts
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 muc=
h,
to bring about a recognition of the problem as one demanding study. These t=
wo
authors are Poincaré and Mach, Poincaré especially in his Science and
Hypothesis, Mach especially in his Analysis of Sensations. Both of them, ho=
wever,
admirable as their work is, seem to me to suffer from a general philosophic=
al
bias. Poincaré is Kantian, while Mach is ultra-empiricist; with Poincaré al=
most
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, dese=
rve
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 considerat=
ions
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 solve=
d 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 havi=
ng
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 s=
ome
repetition of previous definitions.
A relation is sai=
d 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, ag=
ain,
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 relatio=
ns
mentioned just now are also transitive--provided, in the case of "brot=
her
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," &quo=
t;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 posses=
sion
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; hen=
ce
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 exis=
ting
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 symmetri=
cal 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 t=
he
class, while any other common property may be illusory, it is prudent, in o=
rder
to avoid needless assumptions, to substitute the class for the common prope=
rty
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 de=
ny 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, broa=
dly
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 instant=
s.
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 cou=
ld
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 poi=
nts
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 probl=
ems
in connection with this simpler or at least more familiar hypothesis.
The argument agai=
nst
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--wh=
ich
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 feelin=
g is
due, I believe, to a failure to realise imaginatively, as well as abstractl=
y,
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 st=
ill
required in order to feel it: it is necessary to dwell upon it, to thrust o=
ut
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 t=
he
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 theo=
ry
of continuity is in its philosophically important essentials. The applicati=
on
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 theo=
ry
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 continui=
ty
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 t=
he
other. Numbers in order of magnitude, the points on a line from left to rig=
ht,
the moments of time from earlier to later, are instances of series. The not=
ion
of order, which is here introduced, is one which is not required in the the=
ory
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 wiv=
es:
we can see that there must be the same number of husbands as of wives, with=
out
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 se=
t of
terms in themselves, but only to a set in a certain order. A set of terms w=
hich
can be arranged in one order can always also be arranged in other orders, a=
nd 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 ha=
ve
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. Th=
us
between any two fractions, however little they differ, there are an infinite
number of other fractions. Mathematical space and time also have this prope=
rty
of compactness, though whether actual space and time have it is a further q=
uestion,
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 rea=
lise
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 numbe=
r of
terms between any two given terms must be infinite. But when these difficul=
ties
have been solved, the mere compactness in itself offers no great obstacle to
the imagination. In more concrete cases, however, such as motion, compactne=
ss
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 mo=
tion
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 proble=
ms
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 simpl=
ify
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 mathematic=
ian
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 positi=
ons
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 inst=
ant
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 bet=
ween
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 foll=
ows
that our speck must, so long as it moves, pass from one point at one instan=
t 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 fal=
se, 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 no=
t be
supposed to consist in a body's occupying consecutive positions at consecut=
ive
times.
[18] The above paradox is essentially t=
he
same as Zeno's argument of the st=
adium
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 t=
hen
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 o=
f 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 al=
ways
finite distances. If our original distance was an inch, we reach successive=
ly
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 th=
is
kind of question, to fall into an elementary logical blunder. Given any fin=
ite
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 occupi=
es 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 f=
rom
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, l=
ike
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 a=
nd
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, howev=
er
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 compa=
ct
series, and demands, for its full comprehension, that compact series should
have become familiar and easy to the imagination as well as to deliberate t=
hought.
[19] See next lecture.
What is required =
may
be expressed in mathematical language by saying that the position of a movi=
ng
body must be a continuous function of the time. To define accurately what t=
his
means, we proceed as follows. Consider a particle which, at the moment t, i=
s 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 insta=
nts
t1, t2, one earlier than t and one later, such that throughout the whole ti=
me
from t1 to t2 (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 con=
tinuous.
It is obvious that if the particle were to jump suddenly from P to some oth=
er
point Q, our definition would fail for all intervals P1P2 which were too sm=
all
to include Q. Thus our definition affords an analysis of the continuity of
motion, while admitting points and instants and denying infinitesimal dista=
nces
in space or periods in time.
P1
P P2 Q ------|----|----|----|------>
Philosophers, mos=
tly
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 defini=
te
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 whi=
ch
mathematics would lead us to expect, namely by observing a change of positi=
on
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 mig=
ht
apply admirably to some other world; but in this actual world, actual motio=
ns
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 fu=
lly
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 mathematic=
al
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 f=
irst
in one place and then in another. This seems to involve our seeing it simul=
taneously
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 separa=
te
states imagined by the mathematician.
To this objection
there are three supplementary answers, physiological, psychological, and
logical. We will consider them successively.
(1) The physiolog=
ical
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 instan=
taneously
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 phys=
ical
phenomenon: we continue to see it for a few moments after the light-waves h=
ave
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 w=
here
it is at that instant. Sensations, however, as they die away, grow graduall=
y 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 b=
ody
simultaneously, but we shall see them with different degrees of intensity--=
the
present position most vividly, and the others with diminishing vividness, u=
ntil
sensation fades away into immediate memory. This state of things accounts f=
ully
for the perception of motion. A motion is perceived, not merely inferred, w=
hen
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 le=
ss
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 t=
he
immediate object of sense, is assuming the truth of physics, and is thus on=
ly
capable of showing the physical account to be possible, not of showing it t=
o be
necessary. This consideration brings us to the psychological answer.
(2) The psycholog=
ical
answer to our difficulty about motion is part of a vast theory, not yet wor=
ked
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 sketc=
h 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 g=
iven
in sensation, we find it apparently very different from the world of physic=
s.
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 themsel=
ves
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 proposit=
ions
of physics can be translated, by a sort of dictionary, into propositions ab=
out
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 conson=
ant
with the facts than any other equally simple view, to distinguish instantan=
eous
states of objects, and to regard such states as forming a compact series. L=
et
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 o=
ne
sensation. Then, in spite of the fact that we see a finite extent of the mo=
tion
at one instant, the extent which we see at one instant is different from th=
at
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 t=
he
former physical series of points. In fact, though the terms of the series s=
eem
different, the mathematical character of the series is unchanged, and the w=
hole
mathematical theory of motion will apply to it verbatim.
When we are
considering the actual data of sensation in this connection, it is importan=
t 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 thi=
rd,
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 sma=
ll
extra weight may be added, and still no change will be perceived; but if bo=
th
extra weights had been added at once, it may be that the change would be qu=
ite
easily perceptible. Or, again, take shades of colour. It would be easy to f=
ind
three stuffs of such closely similar shades that no difference could be
perceived between the first and second, nor yet between the second and thir=
d,
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 f=
rom the
first. It must, therefore, though indistinguishable from both, be really
intermediate between them.
[21] "Le continu mathématique,&quo=
t;
Revue de Métaphysique et de Morale, vol. i. p. 29.
Such consideratio=
ns
as the above show that, although we cannot distinguish sense-data unless th=
ey
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 po=
int
of view against the mathematical theory of motion are not, therefore,
objections to this theory properly understood, but only to a quite unnecess=
ary
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 i=
nstant
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 numbe=
r of
distinct questions which are apt to be confused when the mathematical conti=
nuum
is said to be inadequate to the facts of sense. We may state these, in orde=
r of
diminishing generality, as follows:--
(a) Are series possessing mathematical continuity logically possible?<= o:p>
(b) Assuming that they are possible
logically, are they not impossi=
ble 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 mathema=
tical
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 t=
hese
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 im=
mediate
experience, the sensible flux is devoid of divisions, and is falsified by t=
he
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, t=
here
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 t=
he
difference of colour in two very neighbouring portions is imperceptible, wh=
ile
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 discre=
te
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 reduc=
tio
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 da=
tum.
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 sen=
se, does
not, theoretically at least, imply even the smallest "knowledge about,=
"
i.e. it does not imply knowledge of any proposition concerning the object w=
ith
which we are acquainted. It is a mistake to speak as if acquaintance had de=
grees:
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 witho=
ut
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 t=
opic
would lead us too far from our theme, and I shall therefore say no more abo=
ut it
at present.
(c) It is sometim=
es
urged that the mathematical account of motion is rendered fictitious by its
assumption of points and instants. Now there are here two different questio=
ns
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 w=
ith
the facts and with logic? (β) is it necessitated by the facts or by lo=
gic?
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, mathema=
tics
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 se=
e,
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 we=
ll
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 the=
ory;
for in practice the refusal to assume points and instants has the same effe=
ct
as the denial of them. But in strict theory the two are quite different, si=
nce
the denial introduces an element of unverifiable dogma which is wholly abse=
nt
when we merely refrain from the assertion. Thus, although we shall derive
points and instants from things, we shall leave the bare possibility open t=
hat
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 compos=
ed
of elements without extension or duration, i.e. of elements which only occu=
py a
point and an instant. Physics, formally, assumes in its differential equati=
ons
that things consist of elements which occupy only a point at each instant, =
but
persist throughout time. For reasons explained in Lecture IV., the persiste=
nce 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 physi=
cs,
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 a=
nd
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 agre=
e 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 n=
ow
face the question: Is there, in actual empirical fact, any sufficient reaso=
n 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 consist=
ent
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 obje=
cts
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 percepti=
on
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 continuo=
us
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--perf=
ectly
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 defin=
ed
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 wor=
ld
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 se=
nse-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 sup=
pose
that an infinite number of sense-data overlap spatially with any given
sense-datum. This hypothesis is possible, if we suppose a single sense-datu=
m,
e.g. in sight, to be a finite surface, enclosing other surfaces which are a=
lso
single sense-data. But there are difficulties in such a hypothesis, and I d=
o 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 low=
er
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 t=
he
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 ot=
her
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 meani=
ng.
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 alte=
red
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, b=
ut
it is a causal psychological fact, not a logical fact. The doctrine would r=
equire
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 concern=
ing
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 philosopher=
s 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 falseh=
ood
is evident as soon as it is stated. The discussion of this question, howeve=
r,
involves so many logical subtleties, and is so beset with difficulties, tha=
t I
shall not pursue it further at present.
When once the abo=
ve
general doctrine is rejected, it is obvious that, where there is change, th=
ere
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 lo=
ng
as we deal with finite periods of time, however short. We are thus driven b=
ack,
by the logical necessities of the case, to the conception of instants witho=
ut
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 ki=
nd
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 whi=
ch
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 enum=
erated
the grounds upon which the reality of the sensible world has been questione=
d,
one of those mentioned was the supposed impossibility of infinity and
continuity. In view of our earlier discussion of physics, it would seem tha=
t 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 continu=
ous
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 illustr=
ated
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." K=
ant
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 fini=
te
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 re=
gards
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 infinit=
y 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 existe=
nce;
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 wh=
ich
are infinite are given all at once by the defining property of their member=
s,
so that there is no question of "completion" or of "successi=
ve
synthesis." And the word "synthesis," by suggesting the ment=
al
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 pr=
oves
is, at most, that if the world had no beginning, it must have already exist=
ed
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 successivel=
y,
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 s=
eries
of events up to the present has an end, since it ends with the present. Owi=
ng
to the inveterate subjectivism of his mental habits, he failed to notice th=
at 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 t=
he
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 reason=
ing.
The second antino=
my
illustrates the dependence of the problem of continuity upon that of infini=
ty.
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 f=
or
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.&quo=
t;
This is like Bergson's objection to "the absurd proposition that motio=
n is
made up of immobilities." Kant does not tell us why he holds that a sp=
ace
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 vi=
ew
is not scientifically or logically necessary, it remains primâ facie possib=
le,
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 impossi=
ble
that space should be composed of points?
I think two
considerations probably influenced him. In the first place, the essential t=
hing
about space is spatial order, and mere points, by themselves, will not acco=
unt
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 reduce=
d 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 rele=
vant
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 on=
ce
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 giv=
es
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 wer=
e.
But the above two grounds, which we have seen to be fallacious, seem suffic=
ient
to account for his opinion, and we may therefore conclude that the antithes=
is
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 objec=
ts
of sense. In the remainder of the present lecture, I wish to state and expl=
ain
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 sha=
ll
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 convinc=
es
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 appea=
rs
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 numbe=
r to
geometry, adopted in that science more arithmetical methods than those with
which Euclid has made us familiar. They, or their contemporaries the atomis=
ts,
believed, apparently, that space is composed of indivisible points, while t=
ime
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 broug=
ht
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 th=
at 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, i=
t is
easy to suppose that there are no other numbers. But this supposition, natu=
ral
as it is, is mistaken.
[22] In what concerns the early Greek
philosophers, my knowledge is lar=
gely
derived from Burnet's valuable work, Early Greek Philosophy (2nd ed., London, 1908). I have also be=
en
greatly assisted by Mr D. S. Robe=
rtson
of Trinity College, who has supplied the deficiencies of my knowledge of Greek, and brought importa=
nt
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 indivisib=
le
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 ent=
ers
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 we=
re 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 di=
fferentiates
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 fou=
rfold
division of mathematical science, attributing one of its parts to the how many, τ
[25] Referred to by Burnet, op. cit., p=
. 120.
[26] iv., 6. 213b, 22; H. Ritter and L.
Preller, Historia Philosophiæ Græ=
cæ,
8th ed., Gotha, 1898, p. 75 (this work will be referred to in future as "R. P.").
This seems to imp=
ly
that they regarded matter as consisting of atoms with empty space in betwee=
n.
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 wh=
ich
beset the Pythagoreans in their attempts to apply numbers arose through the=
ir
discovery of incommensurables, and this, in turn, arose as follows. Pythago=
ras,
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 t=
he
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, th=
ough
it has remained his chief claim to immortality, was soon found to have a co=
nsequence
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 le=
ngth
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, i=
t 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 rat=
io 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 m2 =3D =
2n2. Now
the square of an odd number is odd, but m2, being equal to 2n2, is even. Hence m must be =
even.
But the square of an even number
divides by 4, therefore n2, which is half of m2, 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 mig=
ht
have been assimilated by some philosophies without any great difficulty, bu=
t 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 thou=
ght,
by assuming that he regarded the length of a line as determined by the numb=
er
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, mu=
st 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 s=
ea
for impiously disclosing the terrible discovery to their enemies. It must b=
e 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 in=
to
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 geom=
etry
on a basis which did not assume the universal possibility of numerical meas=
urement--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 rati=
os
of incommensurable lengths. But although irrational numbers have long been =
used
without a qualm, it is only in quite recent years that logically satisfacto=
ry
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 i=
s 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 r=
each
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 fut=
ile
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 Pythagor=
eans,
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 Parme=
nides
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 ch=
ange
follows from this principle; for what is past can be spoken of, and therefo=
re, by
the principle, still is.
[29] So Plato makes Zeno say in the
Parmenides, apropos of his philos=
ophy
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 concept=
ion
of a reality behind the passing illusions of sense, a reality one, indivisi=
ble,
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 log=
ical
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 i=
s 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 tim=
e is
the gate of wisdom. But unimportance is not unreality; and therefore what w=
e shall
have to say about Zeno's arguments in support of Parmenides must be mainly
critical.
The relation of Z=
eno
to Parmenides is explained by Plato[31] in the dialogue in which Socrates, =
as a
young man, learns logical acumen and philosophic disinterestedness from the=
ir
dialectic. I quote from Jowett's translation:
"I see,
Parmenides, said Socrates, that Zeno is your second self in his writings to=
o;
he puts what you say in another way, and would fain deceive us into believi=
ng
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, Socrat=
es,
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 contradicto=
ry
results which they suppose to follow from the affirmation of the one. My an=
swer
is an address to the partisans of the many, whose attack I return with inte=
rest
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 argum=
ents
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 st=
ated
by opponents will realise that a just or adequate presentation of Zeno's po=
sition
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 G=
rèce,
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 supp=
ort 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 premis=
ses,
it is possible to deduce consequences which his opponents must deny. In ord=
er
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 aime=
d at
the Pythagoreans,[34] while others have held that they were intended to ref=
ute
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 Tann=
ery's
opinion, Vorlesungen, 3rd ed. (vo=
l. i.
p. 200).
[36] "Le mouvement et les partisan=
s des
indivisibles," Revue de Métaphysique et de Morale, vol. i. pp.
382-395.
[37] "Le mouvement et les argument=
s 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 adop=
ted
by the Pythagoreans or not, was certainly much advocated, as may be seen fr=
om
the treatise On Indivisible Lines attributed to Aristotle. As regards the f=
irst
two arguments, they would seem to be valid on the hypothesis of indivisible=
s,
and also, without this hypothesis, to be such as would be valid if the
traditional contradictions in infinite numbers were insoluble, which they a=
re
not.
[38] Cf. M. Brochard, "Les prétend=
us
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 wh=
ich
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 t=
he
infinite was infected with contradictions. In a fragment which is not one of
the four famous arguments against motion, Zeno says:
"If things a=
re 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 a=
re 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, wh=
ich
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 yea=
rs
to answer, and which even now are fatal to the teachings of most philosophe=
rs.
The first of Zeno=
's
arguments is the argument of the race-course, which is paraphrased by Burne=
t as
follows:[40]
"You cannot =
get
to the end of a race-course. You cannot traverse an infinite number of poin=
ts
in a finite time. You must traverse the half of any given distance before y=
ou
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 f= irst 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 s= pace 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 thi= ngs, either in regard to divisibility,= or in regard to the ends. Now it is not possible to touch things infinite in regard to numb= er 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 inf= inite, [space] in an infinite [time] and not in a finite [time], and we touch infinite things with infin= ite things, not with finite things.&q= uot; 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 th= ing 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."<= o:p>
Zeno appeals here=
, in
the first place, to the fact that any distance, however small, can be halve=
d.
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 touc=
h 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 y=
ou
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, obt=
ained
by always halving what remains of the course, then the points are reached o=
ne
by one, and, though they are infinite in number, they are in fact all reach=
ed
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 whi=
ch
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 divisibilit=
y,
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 lat=
er
than its predecessor; but the sum of an infinite number of finite times mus=
t be
infinite, and therefore the process will never be completed." It is ve=
ry 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 so=
lely
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 be=
yond
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 Zen=
o's
arguments is the one concerning Achilles and the tortoise, which has achiev=
ed
more notoriety than the others. It is paraphrased by Burnet as follows:[43]=
"Achilles wi=
ll
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 s=
econd
is the so-called Achilles. It con=
sists
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 fr=
om
which the pursued has just departed, so that the slower must necessarily be always still more o=
r less
in advance." Phys., vi. 9. 2=
39B
(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 infini=
te
number of instants make up an infinitely long time is not true, and therefo=
re
the conclusion that Achilles will never overtake the tortoise does not foll=
ow.
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 e=
qual
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 statem=
ent
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 t=
he
first two arguments, the view that a finite part of time consists of a fini=
te
series of successive instants seems to be assumed; at any rate the plausibi=
lity
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 sho=
uld
have parts. Thus, suppose we consider a period consisting of a thousand
instants, and suppose the arrow is in flight throughout this period. At eac=
h 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 chang=
e 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 occup=
ied
at the next moment; but in fact there is no next position and no next momen=
t,
and when once this is imaginatively realised, the difficulty is seen to dis=
appear.
The fourth and la=
st
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 ti=
me
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 veloc=
ity
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. Theref=
ore
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-cours=
e as
they proceed with equal velocity in opposite directions, the one row origin=
ally
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 thi=
nks,
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 equ=
al
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 ha=
lf
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 pas=
sed
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 fi=
rst
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 t=
he
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 the=
m 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 t=
he
other row, are respectively opposite to A, A′, and A″. At the v=
ery
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 pa=
ss
C′? It must have been somewhere between the two moments which we supp=
osed
consecutive, and therefore the two moments cannot really have been consecut=
ive.
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 difficu=
lty,
that B must have passed C′ at some time between two consecutive momen=
ts,
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 th=
at
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 t=
hat
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 mot=
ion
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 can=
not
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 numbe=
r 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 b=
egan
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 t=
ime
and infinity which have been constructed from his day to our own. We have s=
een
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 procee=
ded
on this assumption, while the first and second, which were perhaps intended=
to
refute the opposite assumption, were in that case fallacious. We may theref=
ore escape
from his paradoxes either by maintaining that, though space and time do con=
sist
of points and instants, the number of them in any finite interval is infini=
te;
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 nu=
mber
of philosophers have followed him. Many others, like M. Bergson, have prefe=
rred
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 ad=
missible.
And on grounds which are independent of space and time, infinite numbers, a=
nd
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 s=
ays
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 t=
wo
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 diffi=
culties
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 altoget=
her
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. Etymological=
ly,
"infinite" should mean "having no end." But in fact some
infinite series have ends, some have not; while some collections are infini=
te
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 inst=
ants
from the beginning of time to the present moment has one end, but is infini=
te.
Kant, in his first antinomy, seems to hold that it is harder for the past t=
o 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 mu=
st
learn to leave behind him.
The confusions
introduced into the notions of philosophers by the so-called "true&quo=
t;
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 whi=
ch
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, howeve=
r,
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 reach=
ed
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 argum=
ents
of the race-course and the Achilles. Take the race-course: there is the mom=
ent
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 stric=
tly
unending series. Beyond the whole of this series is the moment when he reac=
hes
the goal. Thus there certainly can be something beyond the whole of an unen=
ding
series. But it remains to show that this fact is only what might have been
expected.
The difficulty, l=
ike
most of the vaguer difficulties besetting the mathematical infinite, is der=
ived,
I think, from the more or less unconscious operation of the idea of countin=
g.
If you set to work to count the terms in an infinite collection, you will n=
ever
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 w=
ould
never reach the goal.
But it is not
essential to the existence of a collection, or even to knowledge and reason=
ing
concerning it, that we should be able to pass its terms in review one by on=
e.
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 th=
is
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 hap=
pens
in the case of infinite collections: they may be known by their characteris=
tics
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, bu=
t 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 difficul=
t as
it is to those who cling obstinately to the prejudices instilled by the ari=
thmetic
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 i= n mathematical symbolism. Why, then, it may be said, should the subject be regarded as philosophy rather than as mathematics? This raises a difficult question, pa= rtly concerned with the use of words, but partly also of real importance in understanding the function of philosophy. Every subject-matter, it would se= em, 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 philoso= phy we follow the inverse direction: from the complex and relatively concrete we proceed towards the simple and abstract by means of analysis, seeking, in t= he process, to eliminate the particularity of the original subject-matter, and= to confine our attention entirely to the logical form of the facts concerned.<= o:p>
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 thos=
e 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 dif=
fer,
however, in their manner of treating the general properties in which all po=
ssible
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 th=
at
can be obtained from them by logical analysis.
The difference be=
tween
philosophy and mathematics may be illustrated by our present problem, namel=
y,
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 the=
se
facts to others, simpler, more fundamental, and inherently more fitted to f=
orm the
premisses of the science of arithmetic. The question, "What is a numbe=
r?"
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 n=
ever
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 u=
nity
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 he=
ld
together and closed, and to that extent as a unity." Now there is in s=
uch
definitions a very elementary blunder, of the same kind that would be commi=
tted
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 thin=
gs
have in common, but is not itself a collection of three things. The definit=
ion,
therefore, apart from any other defects, has failed to reach the necessary
degree of abstraction: the number 3 is something more abstract than any col=
lection
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 t=
hat
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 numbe=
rs
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 reach=
ed
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-m=
erchants,
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 th=
at
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 wh=
at
the number of the objects is. But this operation is really a very complicat=
ed
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 ord=
er,
and to repeat them correctly like the letters of the alphabet, without atta=
ching
any meaning to them. Such a child may count correctly from the point of vie=
w of
a grown-up listener, without having any idea of numbers at all. The operati=
on
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 g=
iven
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 use=
d 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 concern=
ed;
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 res=
pects
in which the infinite numbers that are known differ from finite numbers: fi=
rst,
infinite numbers have, while finite numbers have not, a property which I sh=
all
call reflexiveness; secondly, finite numbers have, while infinite numbers h=
ave
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 Can=
tor
it has come to be recognised that, though at first astonishing, it is no mo=
re 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 nu=
mber
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 addi=
ng
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 exa=
mple
is even more surprising. Write the natural numbers 1, 2, 3, 4, ... in the t=
op
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 f=
rom
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 thu=
s:
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&quo=
t;
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 ma=
de
the modern theory of infinity possible.
[49] Phil. Werke, Gerhardt's edition, v=
ol. 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 Simplic=
ius,
and they reason as follows:
"Simp. Here
already arises a Doubt which I think is not to be resolv'd; and that is thi=
s:
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 fou=
nd
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 possib=
ly 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 t=
hose
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 Facto=
rs
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 n=
ow,
if I should ask how many Roots there are, you can't deny but there are as m=
any
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 m=
any
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 pr=
oportion,
as we go on to bigger Numbers, for count to an Hundred you'll find 10 Squar=
es,
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 Mil=
lion
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 Infinit=
e,
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 mechanic=
s 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 Maste=
r, 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 t=
he
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 limi=
t 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 w=
hat
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 abov=
e discussion
starts, that involves a meaning of greater and less which is not arithmetic=
al.
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 ques=
tion
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 w=
hich
characterises the finite numbers, and which is named after the method of pr=
oof
known as "mathematical induction."
Let us first cons=
ider
what is meant by calling a property "hereditary" in a given serie=
s.
Take such a property as being named Jones. If a man is named Jones, so is h=
is
son; we will therefore call the property of being called Jones hereditary w=
ith
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 successo=
r,
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 th=
is 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 belon=
gs
to 100, and it belongs to 102 because it belongs to 101, and so on--where t=
he
"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 propert=
y,
the next number must always also possess it.
It will be seen t=
hat
a hereditary property, though it must belong to all the finite numbers grea=
ter
than a given number possessing the property, need not belong to all the num=
bers
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 numb=
er.
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 prop=
erty
possessed by Adam must belong to all men; and similarly any hereditary prop=
erty
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 firs=
t 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 successiv=
e additions
of 1; in other words, they are all the numbers that can be reached by count=
ing.
But beyond all th=
ese
numbers, there are the infinite numbers, and infinite numbers do not have a=
ll
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 infi=
nite
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 properti=
es
of numbers, which custom had led people to regard as logically necessary, a=
re
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 lo=
gic,
but only our prejudices and mental habits.
The property of b=
eing
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 obta=
ined
by the addition of 1. It follows that each of the natural numbers is increa=
sed 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; he=
nce
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 propertie=
s of
reflexiveness and non-inductiveness, which we have considered as
characteristics of infinite numbers, have not so far been proved to be alwa=
ys
found together. It is known that all reflexive numbers are non-inductive, b=
ut
it is not known that all non-inductive numbers are reflexive. Fallacious pr=
oofs
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 o=
ur
purposes, therefore, it will be convenient to ignore the bare possibility t=
hat
there may be non-inductive non-reflexive numbers, since all known numbers a=
re
either inductive or reflexive.
When infinite num=
bers
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 t=
hat
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 cons=
ider
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 m=
an.
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 publish=
ed
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 proper=
ties
in a series to which I alluded in connection with inductiveness. His defini=
tion
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 ar=
ticles
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 ye=
ars.
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 investigatio=
n of
the definition of number. He proceeds to show the inadequacy of previous
philosophical theories, especially of the "synthetic a priori" th=
eory
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 s=
ame 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 subje=
cts,
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 distingui=
sh
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 subjecti=
ve,
but non-sensible and objective. This conclusion is important, since it appl=
ies
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 assertio=
n by
recognising the world of logic, which is neither mental nor physical.
The fact is, as F=
rege
points out, that no number, not even 1, is applicable to physical things, b=
ut
only to general terms or descriptions, such as "man," "satel=
lite
of the earth," "satellite of Venus." The general term
"man" is applicable to a certain number of objects: there are in =
the
world so and so many men. The unity which philosophers rightly feel to be
necessary for the assertion of a number is the unity of the general term, a=
nd
it is the general term which is the proper subject of number. And this appl=
ies
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 t=
he
general term "earth's satellite." Similarly, 0 is a property of t=
he
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 occurren=
ces.
Instead of speaki=
ng
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 subjec=
t as
the class or collection of objects--i.e. "mankind" in the above i=
nstance--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 instance=
s;
thus the number depends upon the class, not upon the selection of this or t=
hat
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 wh=
ich 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 amenab=
le
to number in spite of being incapable of enumeration.
Frege next asks t=
he
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 Christi=
an
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 severa=
lly,
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 que=
stion:
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 numb=
er
of a given collection as the class of all collections that are similar to i=
t;
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 whi=
ch
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 ca=
ses.
The above definit=
ion
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 c=
ase is
not without importance. In the first place, it will be found that when an i=
dea
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 te=
nds
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 wou=
ld
be possible to frame definitions more nearly in accordance with our unanaly=
sed
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 plac=
e,
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, m=
ust
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 d=
efinition
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 importa=
nt
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 t=
he
growth of familiarity.
There is, however=
, a
certain logical doctrine which may be thought to form an objection to the a=
bove
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 w=
ould
make havoc of a theory which reduces numbers to classes, and of the many ot=
her
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 cla=
sses
are fictions. What the doctrine is, and why it is not destructive, I will t=
ry
briefly to explain.
On account of cer=
tain
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 sente=
nce
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 mome=
nt's
reflection. For example, in the sentence, "Adam is fond of apples,&quo=
t;
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 t=
hat
can be said significantly about a thing can be said significantly about a c=
lass
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 commo=
n 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 judgmen=
t 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 th=
at
there are three things wherever there are two.
[53] Giles, The Civilisation of China (=
Home
University Library), p. 147.
When it is admitt=
ed
that classes are not things, the question arises: What do we mean by statem=
ents
which are nominally about classes? Take such a statement as, "The clas=
s of
people interested in mathematical logic is not very numerous." Obvious=
ly
this reduces itself to, "Not very many people are interested in
mathematical logic." For the sake of definiteness, let us substitute s=
ome
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 a=
ny reference
at all to a "class." In some such way, all statements nominally a=
bout
a class can be reduced to statements about what follows from the hypothesis=
of
anything's having the defining property of the class. All that is wanted,
therefore, in order to render the verbal use of classes legitimate, is a
uniform method of interpreting propositions in which such a use occurs, so =
as
to obtain propositions in which there is no longer any such use. The defini=
tion
of such a method is a technical matter, which Dr Whitehead and I have dealt
with elsewhere, and which we need not enter into on this occasion.[54]
[54] Cf. Principia Mathematica, § 20, a=
nd
Introduction, chapter iii.
If the theory that
classes are merely symbolic is accepted, it follows that numbers are not ac=
tual
entities, but that propositions in which numbers verbally occur have not re=
ally
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 c=
ase
with all the apparent objects of logic and mathematics. Such words as or, n=
ot, if,
there is, identity, greater, plus, nothing, everything, function, and so on,
are not names of definite objects, like "John" or "Jones,&qu=
ot;
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 sho=
ws
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 u=
se of
unpublished work by my friend Lud=
wig
Wittgenstein.
LECTURE VIII - ON THE NOT=
ION
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 a=
re
simpler, more precise, and less infected with logical redundancy. If the wo=
rk
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 us=
ed
to be felt that the parts of geometry which depend upon the axiom of parall=
els 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 degr=
ee
of certainty of any consequence of the premisses will depend upon that of t=
he
most doubtful premiss employed in proving this consequence. Thus analysis i=
nto
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 th=
is
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 b=
y 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; I=
V.,
how the causality which is used in science differs from that of common sens=
e 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 "cau=
sal
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 in=
fer
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 t=
he
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 s=
ee
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 d=
ue
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 t=
he
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&=
quot;
here in the sense of a class of correlated "aspects," as we d=
id in
Lecture III. Each "aspect" will count separately in stating causal laws=
.
But we cannot bec=
ome
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, si=
nce
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 l=
aw
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 firs=
t is
of the kind in question.
It is to be obser=
ved
that what is constant in a causal law is not the object or objects given, n=
or
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, "s=
ame
cause, same effect," which is sometimes said to be the principle of
causality, is much narrower in its scope than the principle which really oc=
curs
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 wh=
ich
is inferred may be uniquely determined by the causal law, or may be only
described in such general terms that many different particulars might satis=
fy
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, scien=
ce
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 coexist=
ence
between the thing given and the thing inferred. When we hear thunder and in=
fer
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 th=
ing
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 i=
s 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 dra=
wn
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.&q=
uot;
Such a statement involves two causal laws, one inferring from a datum somet=
hing
which existed a quarter of an hour ago, the other inferring from the same d=
atum
something which will exist an hour hence.
Often a causal law
involves not one datum, but many, which need not be all simultaneous with e=
ach
other, though their time-relations must be given. The general scheme of a
causal law will be as follows:
"Whenever th=
ings
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 occ=
ur
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 fi=
nite
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 essenti=
al
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 exist=
ence
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 adequa=
te
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 coexisten=
ce,
for example between touch and sight, between certain sensations in the thro=
at
and the sound of one's own voice, and so on. Every such uniformity of seque=
nce
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 a=
lso.
The connection of experienced past uniformity with expectation as to the fu=
ture
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 an=
imal
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 ha=
ve
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 stat=
ed?
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 lightni=
ng
without hearing thunder; and although, in such a case, we suppose that thun=
der
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 t=
he
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 otherw=
ise
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 suppos=
ed.
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 law=
s?
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 hav=
e at
every instant an acceleration compounded of accelerations towards all the o=
ther
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 sy=
stem
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, t=
he 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 wor=
ld,
the evidence for the universality of causal laws is less complete than in t=
he
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 k=
inds
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 suffi=
ce
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 la=
ws
in which both terms are mental.
It will be noticed
that, although we have spoken of causal laws, we have not hitherto introduc=
ed
the word "cause." At this stage, it will be well to say a few wor=
ds
on legitimate and illegitimate uses of this word. The word "cause,&quo=
t;
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, "Ars=
enic
causes death," so long as we are ignorant of the precise process by wh=
ich
the result is brought about. But in a sufficiently advanced science, the wo=
rd
"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 ca=
ses,
it is convenient to be able to speak of the antecedent event as the
"cause" and the subsequent event as the "effect." In th=
is
sense, provided it is realised that the sequence is not necessary and may h=
ave
exceptions, it is still possible to employ the words "cause" and
"effect." It is in this sense, and in this sense only, that we sh=
all
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 cau=
sal
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 everythi=
ng,
both mental and physical, so far as our observation has extended, has happe=
ned
in accordance with causal laws. The law of universal causation, suggested by
these facts, may be enunciated as follows:
"There are s=
uch
invariable relations between different events at the same or different times
that, given the state of the whole universe throughout any finite time, how=
ever
short, every previous and subsequent event can theoretically be determined =
as a
function of the given events during that time."
Have we any reaso=
n 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 gravitat=
ion,
will continue to hold in the future?
Among observed ca=
usal
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 expectati=
ons,
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 sh=
all
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 val=
id,
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 i=
t 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 thin=
g of
a certain kind is associated in a certain way with a thing of a certain oth=
er
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 i=
ncreases,
the probability approaches indefinitely near to certainty." It may wel=
l be
questioned whether this proposition is true; but if we admit it, we can inf=
er
that any characteristic of the whole of the observed past is likely to appl=
y 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 time=
s,
future as well as past; but without this principle, the observed cases of t=
he
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 inferre=
d.
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 give=
n.
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 cou=
ld 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 o=
f a
"causal law" is found to be far from simple. There must necessari=
ly
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 app=
ear
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 rela=
ted
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 previs=
ion
of the effect; hence the effect becomes the "end" at which the ca=
use
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 pur=
ely
"descriptive" view of physics: physics, they say, does not aim at
telling us "why" things happen, but only "how" they hap=
pen.
And if the question "why?" means anything more than the search fo=
r 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 th=
e scientifically
useful part of the traditional notion of "cause." There is theref=
ore
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 intima=
tely
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 fil=
led
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 c=
ould
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 tha=
t 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 c=
ausal
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, a=
nd
believed to be a cause of C; then, if the belief was correct, we have C, ca=
used
by B, and if the belief was incorrect we have disappointment. Regarded pure=
ly
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 eff=
ects
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 ju=
st
as much as the consequences by the desires; but as we cannot (in general) k=
now
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, consider=
ed
scientifically, has none of that analogy with volition which makes us imagi=
ne
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 eve=
nt,
called the effect; the relation being of such a kind that only one event, o=
r at
any rate only one well-defined sort of event, can have the relation to a gi=
ven
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 rest=
riction.
We shall do better to allow the effect to be before the cause or simultaneo=
us
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 st=
op
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 o=
rder
to have anything approaching certainty concerning the effect, it is necessa=
ry
to include many more circumstances in the cause than unscientific common se=
nse
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 gr=
oup
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 relat=
ion
by enlarging the group. Any such constant relation between events of specif=
ied
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 experienc=
e,
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 cau=
sal
group consists of volitions and the consequent bodily acts, though exceptio=
ns
arise (for example) through sudden paralysis. Another very frequent connect=
ion
(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 n=
ot
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, ol=
d 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 t=
hat,
under the influence of a cool analysis, the doubtful questions involved wil=
l be
found to have no such emotional importance as is sometimes thought, since t=
he
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 analys=
is
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, how=
ever
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, w=
e 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 te=
ll
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, tho=
ugh
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 exter=
nal
compulsion? The two questions, as I shall try to show, are entirely distinc=
t,
and we may answer the first in the affirmative without therefore being forc=
ed
to give an affirmative answer to the second.
(1) Are human act=
ions
theoretically predictable from a sufficient number of antecedents? Let us f=
irst
endeavour to give precision to this question. We may state the question thu=
s:
Is there some constant relation between an act and a certain number of earl=
ier
events, such that, when the earlier events are given, only one act, or at m=
ost 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 pa=
st
that it could not possibly have occurred at any earlier time, and is theref=
ore
necessarily quite different from all previous and subsequent events. If, for
example, I read a certain poem many times, my experience on each occasion i=
s 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, t=
his
principle does not apply to mental events. What is apparently the same caus=
e,
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 i=
t by
which we could imagine it. And on this ground he regards the freedom of the
will as unassailable.
Bergson's content=
ion
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 fore=
see
the whole particularity of the act which will be performed. If he could for=
esee
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 min=
d in
committing the murder, nor whether the murder was to be performed with a kn=
ife
or with a revolver. If the kind of act which will be performed can be fores=
een
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 grous=
e in
the gun-room is told, there will be slight differences due to increasing ha=
bitualness,
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 wh=
at
kind of act will be performed.
Again, his statem=
ent
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 t=
hat
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 wh=
ich
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 possibl=
e,
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 distanc=
e.
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 cau=
ses
whose effects have been observed.
Another respect in
which Bergson's statement of causation is inadequate is in its assumption t=
hat
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 eve=
nts
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 emphatic=
ally
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 t=
he
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. T=
hus
the kinds of cases upon which Bergson relies are insufficient to show the i=
mpossibility
of prediction in the only sense in which prediction has practical or emotio=
nal
interest. We may therefore leave the consideration of his arguments and add=
ress
ourselves to the problem directly.
The law of causat=
ion,
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 categ=
ory
without which science would be impossible. These claims seem to me excessiv=
e.
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 assumptio=
n as
to its truth in other fields. We cannot, therefore, feel any a priori certa=
inty
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 mat=
ter
in the world, and therefore of all the brains and living organisms, the sta=
te
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 th=
at, even
if we admit the most extreme claims of determinism and of correlation of mi=
nd
and brain, still the consequences inimical to what is worth preserving in f=
ree
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 co=
mpel
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 shee=
r mistake.
But this brings us to the second of the two questions which we raised in re=
gard
to free will, namely, whether, assuming determinism, our actions can be in =
any
proper sense regarded as compelled by outside forces.
(2) Are human act=
ions
subject to an external compulsion? We have, in deliberation, a subjective s=
ense
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 hist=
ory.
The supposed inconsistency of these two springs from the habit of conceiving
causes as analogous to volitions--a habit which often survives unconsciousl=
y 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 cause=
s.
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 w=
hat
it was; yet it renders it necessary as a consequence of the data, in the on=
ly
sense in which effects are rendered necessary by their causes. The differen=
ce
which we feel, in this respect, between causes and effects is a mere confus=
ion
due to the fact that we remember past events but do not happen to have memo=
ry of
the future.
The apparent
indeterminateness of the future, upon which some advocates of free will rel=
y,
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. N=
ow,
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 even=
ts
in the same immediate way in which we see past events, what kind of free wi=
ll
would still be possible? Such a kind would be wholly independent of
determinism: it could not be contrary to even the most entirely universal r=
eign
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 be=
ings
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 decisi=
on
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 sati=
sfy
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 volit=
ions
were going to be. Freedom, in short, in any valuable sense, demands only th=
at
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. Everythi=
ng
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, therefor=
e,
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 sai=
d on
philosophical method in the foregoing lectures has been rather by means of
illustrations in particular cases than by means of general precepts. Nothin=
g of
any value can be said on method except through examples; but now, at the en=
d of
our course, we may collect certain general maxims which may possibly be a h=
elp
in acquiring a philosophical habit of mind and a guide in looking for solut=
ions
of philosophic problems.
Philosophy does n=
ot
become scientific by making use of other sciences, in the kind of way in wh=
ich
(e.g.) Herbert Spencer does. Philosophy aims at what is general, and the
special sciences, however they may suggest large generalisations, cannot ma=
ke
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 no=
t be
such as some other science might conceivably contradict. Prophecies as to t=
he
future of the universe, for example, are not the business of philosophy;
whether the universe is progressive, retrograde, or stationary, it is not f=
or
the philosopher to say.
In order to becom=
e a
scientific philosopher, a certain peculiar mental discipline is required. T=
here
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 seem=
s no
hope of its finding any satisfaction. The desire to know philosophical trut=
h 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 desi=
re
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 eff=
orts
to find objections to it, we may obtain the comfort of believing it, althou=
gh,
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 obscur=
ed,
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 unt=
il
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 invent=
ed a
system which fitted admirably with all the facts he knew, except the incomm=
ensurability
of the diagonal of a square and the side; this one little fact stood out, a=
nd
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 i=
mmortality,
while his system has become a matter of merely historical curiosity.[57] Lo=
ve
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 agreea=
ble
results, of whatever kind, has of course been the chief obstacle to honest
philosophising. So strangely perverted do men become by unrecognised passio=
ns,
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 arr=
ive
at an agreeable result than to wish to arrive at a true result. But only th=
ose in
whom the desire to arrive at a true result is paramount can hope to serve a=
ny
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 methodologic=
al
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 intellectu=
al
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 knowle=
dge
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 adop=
tion
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 unl=
ock
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 mathe=
matics
gives the method in physics. And as physics, which, from Plato to the
Renaissance, was as unprogressive, dim, and superstitious as philosophy, be=
came
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 t=
heir
own account, but of quite immeasurable interest as examples of real knowled=
ge
and of a new method whose future fruitfulness he himself divined. But his f=
ew
facts sufficed to destroy the whole vast system of supposed knowledge hande=
d 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 a=
ll
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 ki=
nd
of modesty. It has made the mistake of attacking the interesting problems at
once, instead of proceeding patiently and slowly, accumulating whatever sol=
id
knowledge was obtainable, and trusting the great problems to the future. Me=
n of
science are not ashamed of what is intrinsically trivial, if its consequenc=
es
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 desirabl=
e 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 depe=
nd
upon a number of component problems, usually more abstract than those of wh=
ich
they are the components. It will generally be found that all our initial da=
ta,
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 gen=
eral
and as free from complexity as possible, before the data can be analysed in=
to
the kind of premisses which philosophy aims at discovering. In this process=
of
analysis, the source of difficulty is tracked further and further back, gro=
wing
at each stage more abstract, more refined, more difficult to apprehend. Usu=
ally
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 want=
ed,
as a rule, is some new effort of logical imagination, some glimpse of a pos=
sibility
never conceived before, and then the direct perception that this possibilit=
y 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 sta=
ge
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 intere=
sted
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 gro=
ws
less prodigal of promises. But to the large and still growing body of men e=
ngaged
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 h=
ave
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, unhampe=
red
by the traditions of the past, and not misled by the literary methods of th=
ose
who copy the ancients in all except their merits.