difficulties of infinite numbers, has been disposed of by the positive
theory of the infinite, which will be considered in Lecture VII. But
there remains a feeling--of the kind that led Zeno to the contention
that the arrow in its flight is at rest--which suggests that points and
instants, even if they are infinitely numerous, can only give a jerky
motion, a succession of different immobilities, not the smooth
transitions with which the senses have made us familiar. This feeling is
due, I believe, to a failure to realise imaginatively, as well as
abstractly, the nature of continuous series as they appear in
mathematics. When a theory has been apprehended logically, there is
often a long and serious labour still required in order to _feel_ it: it
is necessary to dwell upon it, to thrust out from the mind, one by one,
the misleading suggestions of false but more familiar theories, to
acquire the kind of intimacy which, in the case of a foreign language,
would enable us to think and dream in it, not merely to construct
laborious sentences by the help of grammar and dictionary. It is, I
believe, the absence of this kind of intimacy which makes many
philosophers regard the mathematical doctrine of continuity as an
inadequate explanation of the continuity which we experience in the
world of sense.
In the present lecture, I shall first try to explain in outline what the
mathematical theory of continuity is in its philosophically important
essentials. The application to actual space and time will not be in
question to begin with. I do not see any reason to suppose that the
points and instants which mathematicians introduce in dealing with space
and time are actual physically existing entities, but I do see reason to
suppose that the continuity of actual space and time may be more or less
analogous to mathematical continuity. The theory of mathematical
continuity is an abstract logical theory, not dependent for its validity
upon any properties of actual space and time. What is claimed for it is
that, when it is understood, certain characteristics of space and time,
previously very hard to analyse, are found not to present any logical
difficulty. What we know empirically about space and time is
insufficient to enable us to decide between various mathematically
possible alternatives, but these alternatives are all fully intelligible
and fully adequate to the observed facts. For the present, however, it
will be well to forget space and time and the continuity of sensible
change, in order to return to these topics equipped with the weapons
provided by the abstract theory of continuity.
Continuity, in mathematics, is a property only possible to a _series_ of
terms, _i.e._ to terms arranged in an order, so that we can say of any
two that one comes _before_ the other. Numbers in order of magnitude,
the points on a line from left to right, the moments of time from
earlier to later, are instances of series. The notion of order, which is
here introduced, is one which is not required in the theory of cardinal
number. It is possible to know that two classes have the same number of
terms without knowing any order in which they are to be taken. We have
an instance of this in such a case as English husbands and English
wives: we can see that there must be the same number of husbands as of
wives, without having to arrange them in a series. But continuity, which
we are now to consider, is essentially a property of an order: it does
not belong to a set of terms in themselves, but only to a set in a
certain order. A set of terms which can be arranged in one order can
always also be arranged in other orders, and a set of terms which can be
arranged in a continuous order can always also be arranged in orders
which are not continuous. Thus the essence of continuity must not be
sought in the nature of the set of terms, but in the nature of their
arrangement in a series.
Mathematicians have distinguished different degrees of continuity, and
have confined the word "continuous," for technical purposes, to series
having a certain high degree of continuity. But for philosophical
purposes, all that is important in continuity is introduced by the
lowest degree of continuity, which is called "compactness." A series is
called "compact" when no two terms are consecutive, but between any two
there are others. One of the simplest examples of a compact series is
the series of fractions in order of magnitude. Given any two fractions,
however near together, there are other fractions greater than the one
and smaller than the other, and therefore no two fractions are
consecutive. There is no fraction, for example, which is next after 1/2:
if we choose some fraction which is very little greater than 1/2, say
51/100 we can find others, such as 101/200, which are nearer to 1/2.
Thus between any two fractions, however little they differ, there are an
infinite number of other fractions. Mathematical space and time also
have this property of compactness, though whether actual space and time
have it is a further question, dependent upon empirical evidence, and
probably incapable of being answered with certainty.
In the case of abstract objects such as fractions, it is perhaps not
very difficult to realise the logical possibility of their forming a
compact series. The difficulties that might be felt are those of
infinity, for in a compact series the number of terms between any two
given terms must be infinite. But when these difficulties have been
solved, the mere compactness in itself offers no great obstacle to the
imagination. In more concrete cases, however, such as motion,
compactness becomes much more repugnant to our habits of thought. It
will therefore be desirable to consider explicitly the mathematical
account of motion, with a view to making its logical possibility felt.
The mathematical account of motion is perhaps artificially simplified
when regarded as describing what actually occurs in the physical world;
but what actually occurs must be capable, by a certain amount of logical
manipulation, of being brought within the scope of the mathematical
account, and must, in its analysis, raise just such problems as are
raised in their simplest form by this account. Neglecting, therefore,
for the present, the question of its physical adequacy, let us devote
ourselves merely to considering its possibility as a formal statement of
the nature of motion.
In order to simplify our problem as much as possible, let us imagine a
tiny speck of light moving along a scale. What do we mean by saying that
the motion is continuous? It is not necessary for our purposes to
consider the whole of what the mathematician means by this statement:
only part of what he means is philosophically important. One part of
what he means is that, if we consider any two positions of the speck
occupied at any two instants, there will be other intermediate positions
occupied at intermediate instants. However near together we take the two
positions, the speck will not jump suddenly from the one to the other,
but will pass through an infinite number of other positions on the way.
Every distance, however small, is traversed by passing through all the
infinite series of positions between the two ends of the distance.
But at this point imagination suggests that we may describe the
continuity of motion by saying that the speck always passes from one
position at one instant to _the next_ position at _the next_ instant. As
soon as we say this or imagine it, we fall into error, because there is
no _next_ point or _next_ instant. If there were, we should find Zeno's
paradoxes, in some form, unavoidable, as will appear in our next
lecture. One simple paradox may serve as an illustration. If our speck
is in motion along the scale throughout the whole of a certain time, it
cannot be at the same point at two consecutive instants. But it cannot,
from one instant to the next, travel further than from one point to the
next, for if it did, there would be no instant at which it was in the
positions intermediate between that at the first instant and that at the
next, and we agreed that the continuity of motion excludes the
possibility of such sudden jumps. It follows that our speck must, so
long as it moves, pass from one point at one instant to the next point
at the next instant. Thus there will be just one perfectly definite
velocity with which all motions must take place: no motion can be faster
than this, and no motion can be slower. Since this conclusion is false,
we must reject the hypothesis upon which it is based, namely that there
are consecutive points and instants.[18] Hence the continuity of motion
must not be supposed to consist in a body's occupying consecutive
positions at consecutive times.
[18] The above paradox is essentially the same as Zeno's argument of
the stadium which will be considered in our next lecture.
The difficulty to imagination lies chiefly, I think, in keeping out the
suggestion of _infinitesimal_ distances and times. Suppose we halve a
given distance, and then halve the half, and so on, we can continue the
process as long as we please, and the longer we continue it, the smaller
the resulting distance becomes. This infinite divisibility seems, at
first sight, to imply that there are infinitesimal distances, _i.e._
distances so small that any finite fraction of an inch would be greater.
This, however, is an error. The continued bisection of our distance,
though it gives us continually smaller distances, gives us always
_finite_ distances. If our original distance was an inch, we reach
successively half an inch, a quarter of an inch, an eighth, a sixteenth,
and so on; but every one of this infinite series of diminishing
distances is finite. "But," it may be said, "_in the end_ the distance
will grow infinitesimal." No, because there is no end. The process of
bisection is one which can, theoretically, be carried on for ever,
without any last term being attained. Thus infinite divisibility of
distances, which must be admitted, does not imply that there are
distances so small that any finite distance would be larger.
It is easy, in this kind of question, to fall into an elementary logical
blunder. Given any finite distance, we can find a smaller distance; this
may be expressed in the ambiguous form "there is a distance smaller than
any finite distance." But if this is then interpreted as meaning "there
is a distance such that, whatever finite distance may be chosen, the
distance in question is smaller," then the statement is false. Common
language is ill adapted to expressing matters of this kind, and
philosophers who have been dependent on it have frequently been misled
by it.
In a continuous motion, then, we shall say that at any given instant the
moving body occupies a certain position, and at other instants it
occupies other positions; the interval between any two instants and
between any two positions is always finite, but the continuity of the
motion is shown in the fact that, however near together we take the two
positions and the two instants, there are an infinite number of
positions still nearer together, which are occupied at instants that are
also still nearer together. The moving body never jumps from one
position to another, but always passes by a gradual transition through
an infinite number of intermediaries. At a given instant, it is where it
is, like Zeno's arrow;[19] but we cannot say that it is at rest at the
instant, since the instant does not last for a finite time, and there is
not a beginning and end of the instant with an interval between them.
Rest consists in being in the same position at all the instants
throughout a certain finite period, however short; it does not consist
simply in a body's being where it is at a given instant. This whole
theory, as is obvious, depends upon the nature of compact series, and
demands, for its full comprehension, that compact series should have
become familiar and easy to the imagination as well as to deliberate
thought.
[19] See next lecture.
What is required may be expressed in mathematical language by saying
that the position of a moving body must be a continuous function of the
time. To define accurately what this means, we proceed as follows.
Consider a particle which, at the moment _t_, is at the point P. Choose
now any small portion P1P2 of the path of the particle, this portion
being one which contains P. We say then that, if the motion of the
particle is continuous at the time _t_, it must be possible to find two
instants _t_1, _t_2, one earlier than _t_ and one later, such that
throughout the whole time from _t_1 to _t_2 (both included), the
particle lies between P1 and P2. And we say that this must still hold
however small we make the portion P1P2. When this is the case, we say
that the motion is continuous at the time _t_; and when the motion is
continuous at all times, we say that the motion as a whole is
continuous. It is obvious that if the particle were to jump suddenly
from P to some other point Q, our definition would fail for all
intervals P1P2 which were too small to include Q. Thus our definition
affords an analysis of the continuity of motion, while admitting points
and instants and denying infinitesimal distances in space or periods in
time.
P1 P P2 Q
------|----|----|----|------>
Philosophers, mostly in ignorance of the mathematician's analysis, have
adopted other and more heroic methods of dealing with the _prima facie_
difficulties of continuous motion. A typical and recent example of
philosophic theories of motion is afforded by Bergson, whose views on
this subject I have examined elsewhere.[20]
[20] _Monist_, July 1912, pp. 337-341.
Apart from definite arguments, there are certain feelings, rather than
reasons, which stand in the way of an acceptance of the mathematical
account of motion. To begin with, if a body is moving at all fast, we
_see_ its motion just as we see its colour. A _slow_ motion, like that
of the hour-hand of a watch, is only known in the way which mathematics
would lead us to expect, namely by observing a change of position after
a lapse of time; but, when we observe the motion of the second-hand, we
do not merely see first one position and then another--we see something
as directly sensible as colour. What is this something that we see, and
that we call visible motion? Whatever it is, it is _not_ the successive
occupation of successive positions: something beyond the mathematical
theory of motion is required to account for it. Opponents of the
mathematical theory emphasise this fact. "Your theory," they say, "may
be very logical, and might apply admirably to some other world; but in
this actual world, actual motions are quite different from what your
theory would declare them to be, and require, therefore, some different
philosophy from yours for their adequate explanation."
The objection thus raised is one which I have no wish to underrate, but
I believe it can be fully answered without departing from the methods
and the outlook which have led to the mathematical theory of motion. Let
us, however, first try to state the objection more fully.
If the mathematical theory is adequate, nothing happens when a body
moves except that it is in different places at different times. But in
this sense the hour-hand and the second-hand are equally in motion, yet
in the second-hand there is something perceptible to our senses which is
absent in the hour-hand. We can see, at each moment, that the
second-hand _is moving_, which is different from seeing it first in one
place and then in another. This seems to involve our seeing it
simultaneously in a number of places, although it must also involve our
seeing that it is in some of these places earlier than in others. If,
for example, I move my hand quickly from left to right, you seem to see
the whole movement at once, in spite of the fact that you know it begins
at the left and ends at the right. It is this kind of consideration, I
think, which leads Bergson and many others to regard a movement as
really one indivisible whole, not the series of separate states imagined
by the mathematician.
To this objection there are three supplementary answers, physiological,
psychological, and logical. We will consider them successively.
(1) The physiological answer merely shows that, if the physical world is
what the mathematician supposes, its sensible appearance may
nevertheless be expected to be what it is. The aim of this answer is
thus the modest one of showing that the mathematical account is not
impossible as applied to the physical world; it does not even attempt to
show that this account is necessary, or that an analogous account
applies in psychology.
When any nerve is stimulated, so as to cause a sensation, the sensation
does not cease instantaneously with the cessation of the stimulus, but
dies away in a short finite time. A flash of lightning, brief as it is
to our sight, is briefer still as a physical phenomenon: we continue to
see it for a few moments after the light-waves have ceased to strike the
eye. Thus in the case of a physical motion, if it is sufficiently swift,
we shall actually at one instant see the moving body throughout a finite
portion of its course, and not only at the exact spot where it is at
that instant. Sensations, however, as they die away, grow gradually
fainter; thus the sensation due to a stimulus which is recently past is
not exactly like the sensation due to a present stimulus. It follows
from this that, when we see a rapid motion, we shall not only see a
number of positions of the moving body simultaneously, but we shall see
them with different degrees of intensity--the present position most
vividly, and the others with diminishing vividness, until sensation
fades away into immediate memory. This state of things accounts fully
for the perception of motion. A motion is _perceived_, not merely
_inferred_, when it is sufficiently swift for many positions to be
sensible at one time; and the earlier and later parts of one perceived
motion are distinguished by the less and greater vividness of the
sensations.
This answer shows that physiology can account for our perception of
motion. But physiology, in speaking of stimulus and sense-organs and a
physical motion distinct from the immediate object of sense, is assuming
the truth of physics, and is thus only capable of showing the physical
account to be possible, not of showing it to be _necessary_. This
consideration brings us to the psychological answer.
(2) The psychological answer to our difficulty about motion is part of a
vast theory, not yet worked out, and only capable, at present, of being
vaguely outlined. We considered this theory in the third and fourth
lectures; for the present, a mere sketch of its application to our
present problem must suffice. The world of physics, which was assumed in
the physiological answer, is obviously inferred from what is given in
sensation; yet as soon as we seriously consider what is actually given
in sensation, we find it apparently very different from the world of
physics. The question is thus forced upon us: Is the inference from
sense to physics a valid one? I believe the answer to be affirmative,
for reasons which I suggested in the third and fourth lectures; but the
answer cannot be either short or easy. It consists, broadly speaking, in
showing that, although the particles, points, and instants with which
physics operates are not themselves given in experience, and are very
likely not actually existing things, yet, out of the materials provided
in sensation, it is possible to make logical constructions having the
mathematical properties which physics assigns to particles, points, and
instants. If this can be done, then all the propositions of physics can
be translated, by a sort of dictionary, into propositions about the
kinds of objects which are given in sensation.
Applying these general considerations to the case of motion, we find
that, even within the sphere of immediate sense-data, it is necessary,
or at any rate more consonant with the facts than any other equally
simple view, to distinguish instantaneous states of objects, and to
regard such states as forming a compact series. Let us consider a body
which is moving swiftly enough for its motion to be perceptible, and
long enough for its motion to be not wholly comprised in one sensation.
Then, in spite of the fact that we see a finite extent of the motion at
one instant, the extent which we see at one instant is different from
that which we see at another. Thus we are brought back, after all, to a
series of momentary views of the moving body, and this series will be
compact, like the former physical series of points. In fact, though the
_terms_ of the series seem different, the mathematical character of the
series is unchanged, and the whole mathematical theory of motion will
apply to it _verbatim_.
When we are considering the actual data of sensation in this connection,
it is important to realise that two sense-data may be, and _must_
sometimes be, really different when we cannot perceive any difference
between them. An old but conclusive reason for believing this was
emphasised by Poincare.[21] In all cases of sense-data capable of
gradual change, we may find one sense-datum indistinguishable from
another, and that other indistinguishable from a third, while yet the
first and third are quite easily distinguishable. Suppose, for example,
a person with his eyes shut is holding a weight in his hand, and someone
noiselessly adds a small extra weight. If the extra weight is small
enough, no difference will be perceived in the sensation. After a time,
another small extra weight may be added, and still no change will be
perceived; but if both extra weights had been added at once, it may be
that the change would be quite easily perceptible. Or, again, take
shades of colour. It would be easy to find three stuffs of such closely
similar shades that no difference could be perceived between the first
and second, nor yet between the second and third, while yet the first
and third would be distinguishable. In such a case, the second shade
cannot be the same as the first, or it would be distinguishable from the
third; nor the same as the third, or it would be distinguishable from
the first. It must, therefore, though indistinguishable from both, be
really intermediate between them.
[21] "Le continu mathematique," _Revue de Metaphysique et de Morale_,
vol. i. p. 29.
Such considerations as the above show that, although we cannot
distinguish sense-data unless they differ by more than a certain amount,
it is perfectly reasonable to suppose that sense-data of a given kind,
such as weights or colours, really form a compact series. The objections
which may be brought from a psychological point of view against the
mathematical theory of motion are not, therefore, objections to this
theory properly understood, but only to a quite unnecessary assumption
of simplicity in the momentary object of sense. Of the immediate object
of sense, in the case of a visible motion, we may say that at each
instant it is in all the positions which remain sensible at that
instant; but this set of positions changes continuously from moment to
moment, and is amenable to exactly the same mathematical treatment as if
it were a mere point. When we assert that some mathematical account of
phenomena is correct, all that we primarily assert is that _something_
definable in terms of the crude phenomena satisfies our formulae; and in
this sense the mathematical theory of motion is applicable to the data
of sensation as well as to the supposed particles of abstract physics.
There are a number of distinct questions which are apt to be confused
when the mathematical continuum is said to be inadequate to the facts of
sense. We may state these, in order of diminishing generality, as
follows:--
(a) Are series possessing mathematical continuity logically
possible?
(b) Assuming that they are possible logically, are they not
impossible as applied to actual sense-data, because, among actual
sense-data, there are no such fixed mutually external terms as are
to be found, _e.g._, in the series of fractions?
(c) Does not the assumption of points and instants make the whole
mathematical account fictitious?
(d) Finally, assuming that all these objections have been answered,
is there, in actual empirical fact, any sufficient reason to believe
the world of sense continuous?
Let us consider these questions in succession.