continuum turns partly on the elementary misunderstandings we considered
at the beginning of the present lecture, partly on the possibility of
the mathematical infinite, which will occupy our next two lectures, and
partly on the logical form of the answer to the Bergsonian objection
which we stated a few minutes ago. I shall say no more on this topic at
present, since it is desirable first to complete the psychological
answer.
(b) The question whether sense-data are composed of mutually external
units is not one which can be decided by empirical evidence. It is often
urged that, as a matter of immediate experience, the sensible flux is
devoid of divisions, and is falsified by the dissections of the
intellect. Now I have no wish to argue that this view is _contrary_ to
immediate experience: I wish only to maintain that it is essentially
incapable of being _proved_ by immediate experience. As we saw, there
must be among sense-data differences so slight as to be imperceptible:
the fact that sense-data are immediately given does not mean that their
differences also _must_ be immediately given (though they _may_ be).
Suppose, for example, a coloured surface on which the colour changes
gradually--so gradually that the difference of colour in two very
neighbouring portions is imperceptible, while the difference between
more widely separated portions is quite noticeable. The effect produced,
in such a case, will be precisely that of "interpenetration," of
transition which is not a matter of discrete units. And since it tends
to be supposed that the colours, being immediate data, must _appear_
different if they _are_ different, it seems easily to follow that
"interpenetration" must be the ultimately right account. But this does
not follow. It is unconsciously assumed, as a premiss for a _reductio ad
absurdum_ of the analytic view, that, if A and B are immediate data, and
A differs from B, then the fact that they differ must also be an
immediate datum. It is difficult to say how this assumption arose, but I
think it is to be connected with the confusion between "acquaintance"
and "knowledge about." Acquaintance, which is what we derive from sense,
does not, theoretically at least, imply even the smallest "knowledge
about," _i.e._ it does not imply knowledge of any proposition concerning
the object with which we are acquainted. It is a mistake to speak as if
acquaintance had degrees: there is merely acquaintance and
non-acquaintance. When we speak of becoming "better acquainted," as for
instance with a person, what we must mean is, becoming acquainted with
more parts of a certain whole; but the acquaintance with each part is
either complete or nonexistent. Thus it is a mistake to say that if we
were perfectly acquainted with an object we should know all about it.
"Knowledge about" is knowledge of propositions, which is not involved
necessarily in acquaintance with the constituents of the propositions.
To know that two shades of colour are different is knowledge about them;
hence acquaintance with the two shades does not in any way necessitate
the knowledge that they are different.
From what has just been said it follows that the nature of sense-data
cannot be validly used to prove that they are not composed of mutually
external units. It may be admitted, on the other hand, that nothing in
their empirical character specially necessitates the view that they are
composed of mutually external units. This view, if it is held, must be
held on logical, not on empirical, grounds. I believe that the logical
grounds are adequate to the conclusion. They rest, at bottom, upon the
impossibility of explaining complexity without assuming constituents. It
is undeniable that the visual field, for example, is complex; and so far
as I can see, there is always self-contradiction in the theories which,
while admitting this complexity, attempt to deny that it results from a
combination of mutually external units. But to pursue this topic would
lead us too far from our theme, and I shall therefore say no more about
it at present.
(c) It is sometimes urged that the mathematical account of motion is
rendered fictitious by its assumption of points and instants. Now there
are here two different questions to be distinguished. There is the
question of absolute or relative space and time, and there is the
question whether what occupies space and time must be composed of
elements which have no extension or duration. And each of these
questions in turn may take two forms, namely: (A) is the hypothesis
_consistent_ with the facts and with logic? (B) is it _necessitated_ by
the facts or by logic? I wish to answer, in each case, yes to the first
form of the question, and no to the second. But in any case the
mathematical account of motion will not be fictitious, provided a right
interpretation is given to the words "point" and "instant." A few words
on each alternative will serve to make this clear.
Formally, mathematics adopts an absolute theory of space and time,
_i.e._ it assumes that, besides the things which are in space and time,
there are also entities, called "points" and "instants," which are
occupied by things. This view, however, though advocated by Newton, has
long been regarded by mathematicians as merely a convenient fiction.
There is, so far as I can see, no conceivable evidence either for or
against it. It is logically possible, and it is consistent with the
facts. But the facts are also consistent with the denial of spatial and
temporal entities over and above things with spatial and temporal
relations. Hence, in accordance with Occam's razor, we shall do well to
abstain from either assuming or denying points and instants. This means,
so far as practical working out is concerned, that we adopt the
relational theory; for in practice the refusal to assume points and
instants has the same effect as the denial of them. But in strict theory
the two are quite different, since the denial introduces an element of
unverifiable dogma which is wholly absent when we merely refrain from
the assertion. Thus, although we shall derive points and instants from
things, we shall leave the bare possibility open that they may also have
an independent existence as simple entities.
We come now to the question whether the things in space and time are to
be conceived as composed of elements without extension or duration,
_i.e._ of elements which only occupy a point and an instant. Physics,
formally, assumes in its differential equations that things consist of
elements which occupy only a point at each instant, but persist
throughout time. For reasons explained in Lecture IV., the persistence
of things through time is to be regarded as the formal result of a
logical construction, not as necessarily implying any actual
persistence. The same motives, in fact, which lead to the division of
things into point-particles, ought presumably to lead to their division
into instant-particles, so that the ultimate _formal_ constituent of the
matter in physics will be a point-instant-particle. But such objects, as
well as the particles of physics, are not data. The same economy of
hypothesis, which dictates the practical adoption of a relative rather
than an absolute space and time, also dictates the practical adoption of
material elements which have a finite extension and duration. Since, as
we saw in Lecture IV., points and instants can be constructed as logical
functions of such elements, the mathematical account of motion, in which
a particle passes continuously through a continuous series of points,
can be interpreted in a form which assumes only elements which agree
with our actual data in having a finite extension and duration. Thus, so
far as the use of points and instants is concerned, the mathematical
account of motion can be freed from the charge of employing fictions.
(d) But we must now face the question: Is there, in actual empirical
fact, any sufficient reason to believe the world of sense continuous?
The answer here must, I think, be in the negative. We may say that the
hypothesis of continuity is perfectly consistent with the facts and with
logic, and that it is technically simpler than any other tenable
hypothesis. But since our powers of discrimination among very similar
sensible objects are not infinitely precise, it is quite impossible to
decide between different theories which only differ in regard to what is
below the margin of discrimination. If, for example, a coloured surface
which we see consists of a finite number of very small surfaces, and if
a motion which we see consists, like a cinematograph, of a large finite
number of successive positions, there will be nothing empirically
discoverable to show that objects of sense are not continuous. In what
is called _experienced_ continuity, such as is said to be given in
sense, there is a large negative element: absence of perception of
difference occurs in cases which are _thought_ to give perception of
absence of difference. When, for example, we cannot distinguish a colour
A from a colour B, nor a colour B from a colour C, but can distinguish A
from C, the indistinguishability is a purely negative fact, namely, that
we do not _perceive_ a difference. Even in regard to immediate data,
this is no reason for denying that there is a difference. Thus, if we
see a coloured surface whose colour changes gradually, its sensible
appearance if the change is continuous will be indistinguishable from
what it would be if the change were by small finite jumps. If this is
true, as it seems to be, it follows that there can never be any
empirical evidence to demonstrate that the sensible world is continuous,
and not a collection of a very large finite number of elements of which
each differs from its neighbour in a finite though very small degree.
The continuity of space and time, the infinite number of different
shades in the spectrum, and so on, are all in the nature of unverifiable
hypotheses--perfectly possible logically, perfectly consistent with the
known facts, and simpler technically than any other tenable hypotheses,
but not the sole hypotheses which are logically and empirically
adequate.
If a relational theory of instants is constructed, in which an "instant"
is defined as a group of events simultaneous with each other and not all
simultaneous with any event outside the group, then if our resulting
series of instants is to be compact, it must be possible, if _x_ wholly
precedes _y_, to find an event _z_, simultaneous with part of _x_, which
wholly precedes some event which wholly precedes _y_. Now this requires
that the number of events concerned should be infinite in any finite
period of time. If this is to be the case in the world of one man's
sense-data, and if each sense-datum is to have not less than a certain
finite temporal extension, it will be necessary to assume that we always
have an infinite number of sense-data simultaneous with any given
sense-datum. Applying similar considerations to space, and assuming that
sense-data are to have not less than a certain spatial extension, it
will be necessary to suppose that an infinite number of sense-data
overlap spatially with any given sense-datum. This hypothesis is
possible, if we suppose a single sense-datum, _e.g._ in sight, to be a
finite surface, enclosing other surfaces which are also single
sense-data. But there are difficulties in such a hypothesis, and I do
not know whether these difficulties could be successfully met. If they
cannot, we must do one of two things: either declare that the world of
one man's sense-data is not continuous, or else refuse to admit that
there is any lower limit to the duration and extension of a single
sense-datum. I do not know what is the right course to adopt as regards
these alternatives. The logical analysis we have been considering
provides the apparatus for dealing with the various hypotheses, and the
empirical decision between them is a problem for the psychologist.
(3) We have now to consider the _logical_ answer to the alleged
difficulties of the mathematical theory of motion, or rather to the
positive theory which is urged on the other side. The view urged
explicitly by Bergson, and implied in the doctrines of many
philosophers, is, that a motion is something indivisible, not validly
analysable into a series of states. This is part of a much more general
doctrine, which holds that analysis always falsifies, because the parts
of a complex whole are different, as combined in that whole, from what
they would otherwise be. It is very difficult to state this doctrine in
any form which has a precise meaning. Often arguments are used which
have no bearing whatever upon the question. It is urged, for example,
that when a man becomes a father, his nature is altered by the new
relation in which he finds himself, so that he is not strictly identical
with the man who was previously not a father. This may be true, but it
is a causal psychological fact, not a logical fact. The doctrine would
require that a man who is a father cannot be strictly identical with a
man who is a son, because he is modified in one way by the relation of
fatherhood and in another by that of sonship. In fact, we may give a
precise statement of the doctrine we are combating in the form: _There
can never be two facts concerning the same thing._ A fact concerning a
thing always is or involves a relation to one or more entities; thus two
facts concerning the same thing would involve two relations of the same
thing. But the doctrine in question holds that a thing is so modified by
its relations that it cannot be the same in one relation as in another.
Hence, if this doctrine is true, there can never be more than one fact
concerning any one thing. I do not think the philosophers in question
have realised that this is the precise statement of the view they
advocate, because in this form the view is so contrary to plain truth
that its falsehood is evident as soon as it is stated. The discussion of
this question, however, involves so many logical subtleties, and is so
beset with difficulties, that I shall not pursue it further at present.
When once the above general doctrine is rejected, it is obvious that,
where there is change, there must be a succession of states. There
cannot be change--and motion is only a particular case of change--unless
there is something different at one time from what there is at some
other time. Change, therefore, must involve relations and complexity,
and must demand analysis. So long as our analysis has only gone as far
as other smaller changes, it is not complete; if it is to be complete,
it must end with terms that are not changes, but are related by a
relation of earlier and later. In the case of changes which appear
continuous, such as motions, it seems to be impossible to find anything
other than change so long as we deal with finite periods of time,
however short. We are thus driven back, by the logical necessities of
the case, to the conception of instants without duration, or at any rate
without any duration which even the most delicate instruments can
reveal. This conception, though it can be made to seem difficult, is
really easier than any other that the facts allow. It is a kind of
logical framework into which any tenable theory must fit--not
necessarily itself the statement of the crude facts, but a form in which
statements which are true of the crude facts can be made by a suitable
interpretation. The direct consideration of the crude facts of the
physical world has been undertaken in earlier lectures; in the present
lecture, we have only been concerned to show that nothing in the crude
facts is inconsistent with the mathematical doctrine of continuity, or
demands a continuity of a radically different kind from that of
mathematical motion.
LECTURE VI
THE PROBLEM OF INFINITY CONSIDERED HISTORICALLY
It will be remembered that, when we enumerated the grounds upon which
the reality of the sensible world has been questioned, one of those
mentioned was the supposed impossibility of infinity and continuity. In
view of our earlier discussion of physics, it would seem that no
_conclusive_ empirical evidence exists in favour of infinity or
continuity in objects of sense or in matter. Nevertheless, the
explanation which assumes infinity and continuity remains incomparably
easier and more natural, from a scientific point of view, than any
other, and since Georg Cantor has shown that the supposed contradictions
are illusory, there is no longer any reason to struggle after a finitist
explanation of the world.
The supposed difficulties of continuity all have their source in the
fact that a continuous series must have an infinite number of terms, and
are in fact difficulties concerning infinity. Hence, in freeing the
infinite from contradiction, we are at the same time showing the logical
possibility of continuity as assumed in science.
The kind of way in which infinity has been used to discredit the world
of sense may be illustrated by Kant's first two antinomies. In the
first, the thesis states: "The world has a beginning in time, and as
regards space is enclosed within limits"; the antithesis states: "The
world has no beginning and no limits in space, but is infinite in
respect of both time and space." Kant professes to prove both these
propositions, whereas, if what we have said on modern logic has any
truth, it must be impossible to prove either. In order, however, to
rescue the world of sense, it is enough to destroy the proof of _one_ of
the two. For our present purpose, it is the proof that the world is
_finite_ that interests us. Kant's argument as regards space here rests
upon his argument as regards time. We need therefore only examine the
argument as regards time. What he says is as follows:
"For let us assume that the world has no beginning as regards time, so
that up to every given instant an eternity has elapsed, and therefore an
infinite series of successive states of the things in the world has
passed by. But the infinity of a series consists just in this, that it
can never be completed by successive synthesis. Therefore an infinite
past world-series is impossible, and accordingly a beginning of the
world is a necessary condition of its existence; which was the first
thing to be proved."
Many different criticisms might be passed on this argument, but we will
content ourselves with a bare minimum. To begin with, it is a mistake to
define the infinity of a series as "impossibility of completion by
successive synthesis." The notion of infinity, as we shall see in the
next lecture, is primarily a property of _classes_, and only
derivatively applicable to series; classes which are infinite are given
all at once by the defining property of their members, so that there is
no question of "completion" or of "successive synthesis." And the word
"synthesis," by suggesting the mental activity of synthesising,
introduces, more or less surreptitiously, that reference to mind by
which all Kant's philosophy was infected. In the second place, when Kant
says that an infinite series can "never" be completed by successive
synthesis, all that he has even conceivably a right to say is that it
cannot be completed _in a finite time_. Thus what he really proves is,
at most, that if the world had no beginning, it must have already
existed for an infinite time. This, however, is a very poor conclusion,
by no means suitable for his purposes. And with this result we might, if
we chose, take leave of the first antinomy.
It is worth while, however, to consider how Kant came to make such an
elementary blunder. What happened in his imagination was obviously
something like this: Starting from the present and going backwards in
time, we have, if the world had no beginning, an infinite series of
events. As we see from the word "synthesis," he imagined a mind trying
to grasp these successively, _in the reverse order_ to that in which
they had occurred, _i.e._ going from the present backwards. _This_
series is obviously one which has no end. But the series of events up to
the present has an end, since it ends with the present. Owing to the
inveterate subjectivism of his mental habits, he failed to notice that
he had reversed the sense of the series by substituting backward
synthesis for forward happening, and thus he supposed that it was
necessary to identify the mental series, which had no end, with the
physical series, which had an end but no beginning. It was this mistake,
I think, which, operating unconsciously, led him to attribute validity
to a singularly flimsy piece of fallacious reasoning.
The second antinomy illustrates the dependence of the problem of
continuity upon that of infinity. The thesis states: "Every complex
substance in the world consists of simple parts, and there exists
everywhere nothing but the simple or what is composed of it." The
antithesis states: "No complex thing in the world consists of simple
parts, and everywhere in it there exists nothing simple." Here, as
before, the proofs of both thesis and antithesis are open to criticism,
but for the purpose of vindicating physics and the world of sense it is
enough to find a fallacy in _one_ of the proofs. We will choose for this
purpose the proof of the antithesis, which begins as follows:
"Assume that a complex thing (as substance) consists of simple parts.
Since all external relation, and therefore all composition out of
substances, is only possible in space, the space occupied by a complex
thing must consist of as many parts as the thing consists of. Now space
does not consist of simple parts, but of spaces."
The rest of his argument need not concern us, for the nerve of the proof
lies in the one statement: "Space does not consist of simple parts, but
of spaces." This is like Bergson's objection to "the absurd proposition
that motion is made up of immobilities." Kant does not tell us why he
holds that a space must consist of spaces rather than of simple parts.
Geometry regards space as made up of points, which are simple; and
although, as we have seen, this view is not scientifically or logically
_necessary_, it remains _prima facie_ possible, and its mere possibility
is enough to vitiate Kant's argument. For, if his proof of the thesis of
the antinomy were valid, and if the antithesis could only be avoided by
assuming points, then the antinomy itself would afford a conclusive
reason in favour of points. Why, then, did Kant think it impossible that
space should be composed of points?
I think two considerations probably influenced him. In the first place,
the essential thing about space is spatial order, and mere points, by
themselves, will not account for spatial order. It is obvious that his
argument assumes absolute space; but it is spatial _relations_ that are
alone important, and they cannot be reduced to points. This ground for
his view depends, therefore, upon his ignorance of the logical theory of
order and his oscillations between absolute and relative space. But
there is also another ground for his opinion, which is more relevant to
our present topic. This is the ground derived from infinite
divisibility. A space may be halved, and then halved again, and so on
_ad infinitum_, and at every stage of the process the parts are still
spaces, not points. In order to reach points by such a method, it would
be necessary to come to the end of an unending process, which is
impossible. But just as an infinite class can be given all at once by
its defining concept, though it cannot be reached by successive
enumeration, so an infinite set of points can be given all at once as
making up a line or area or volume, though they can never be reached by
the process of successive division. Thus the infinite divisibility of
space gives no ground for denying that space is composed of points. Kant
does not give his grounds for this denial, and we can therefore only
conjecture what they were. But the above two grounds, which we have seen
to be fallacious, seem sufficient to account for his opinion, and we may
therefore conclude that the antithesis of the second antinomy is
unproved.