OF THE PERCEPTION OF TIME AND NUMBER.
1. OUR perception of the passage of time involves a series
of acts of memory. This is easily seen and assented to, when
large intervals of time and a complex train of occurrences
are concerned. But since memory is requisite in order to
apprehend time in such cases, we cannot doubt that the same
faculty must be concerned in the shortest and simplest cases
of succession; for it will hardly be maintained that the
process by which we contemplate the progress of time is
different, when small, and when large intervals are
concerned. If memory be absolutely requisite to connect two
events which begin and end a day, and to perceive a tract of
time between them, it must be equally indispensable to
connect the beginning and end of a minute, or a second;
though in this case the effort may be smaller, and
consequently more easily overlooked. In common cases, we are
unconscious of the act of thought by which we recollect the
preceding instant, though we perceive the effort when we
recollect some distant event. And this is analogous to what
happens in other instances. Thus, we walk without being
conscious of the volitions by which we move our muscles;
but, in order to leap, a distinct and manifest exertion of
the same muscles is necessary. Yet no one will doubt that we
walk as well as leap by an act of the will exerted through
the muscles; and in like manner, our consciousness of small
as well as large intervals of time involves something of the
nature of an act of memory.
2. But this constant and almost imperceptible kind of
memory, by which we connect the beginning and {142} end of
each instant as it passes, may very fitly be distinguished
in common cases from manifest acts of recollection, although
it may be difficult or impossible to separate the two
operations in general. This perpetual and latent kind of
memory may be termed a _sense of successiveness_; and must
be considered as an internal sense by which we perceive
ourselves existing in time, much in the same way as by our
external and muscular sense we perceive ourselves existing
in space. And both our internal thoughts and feelings, and
the events which take place around us, are apprehended as
objects of this internal sense, and thus as taking place in
time.
3. In the same manner in which our interpretation of the
notices of the muscular sense implies the power of moving
our limbs, and of touching at will this object or that; our
apprehension of the relations of time, by means of the
internal sense of successiveness, implies a power of
recalling what has past, and of retaining what is passing.
We are able to seize the occurrences which have just taken
place, and to hold them fast in our minds so as mentally to
measure their distance in time from occurrences now present.
And thus, this sense of successiveness, like the muscular
sense with which we have compared it, implies activity of
the mind itself, and is not a sense passively receiving
impressions.
4. The conception of _Number_ appears to require the
exercise of the same sense of succession. At first sight,
indeed, we seem to apprehend Number without any act of
memory, or any reference to time: for example, we look at a
horse, and see that his legs are four; and this we seem to
do at once, without reckoning them. But it is not difficult
to see that this seeming instantaneousness of the perception
of small numbers is an illusion. This resembles the many
other cases in which we perform short and easy acts so
rapidly and familiarly that we are unconscious of them; as
in the acts of seeing, and of articulating our words. And
this is the more manifest, since we begin our acquaintance
with number by counting even the {143} smallest numbers.
Children and very rude savages must use an effort to reckon
even their five fingers, and find a difficulty in going
further. And persons have been known who were able by habit,
or by a peculiar natural aptitude, to count by dozens as
rapidly as common persons can by units. We may conclude,
therefore, that when we appear to catch a small number by a
single glance of the eye, we do in fact count the units of
it in a regular, though very brief succession. To count
requires an act of memory. Of this we are sensible when we
count very slowly, as when we reckon the strokes of a
church-clock; for in such a case we may forget in the
intervals of the strokes, and _miscount_. Now it will not be
doubted that the nature of the process in counting is the
same whether we count fast or slow. There is no definite
speed of reckoning at which the faculties which it requires
are changed; and therefore memory, which is requisite in
some cases, must be so in all[13\2].
[Note 13\2: I have considered Number as involving the
exercise of the sense of succession, because I cannot draw
any line between those cases of large numbers, in which, the
process of counting being performed, there is a manifest
apprehension of succession; and those cases of small
numbers, in which we seem to see the number at one glance.
But if any one holds Number to be apprehended by a direct
act of intuition, as Space and Time are, this view will not
disturb the other doctrines delivered in the text.]
The act of counting, (_one_, _two_, _three_, and so on,) is
the foundation of all our knowledge of number. The intuition
of the relations of number involves this act of counting;
for, as we have just seen, the conception of number cannot
be obtained in any other way. And thus the whole of
theoretical arithmetic depends upon an act of the mind, and
upon the conditions which the exercise of that act implies.
These have been already explained in the last chapter.
5. But if the apprehension of number be accompanied by an
act of the mind, the apprehension of _rhythm_ is so still
more clearly. All the forms of versification and the
_measures_ of melodies are the creations of man, who thus
realizes in words and sounds the {144} forms of recurrence
which rise within his own mind. When we hear in a quiet
scene any rapidly-repeated sound, as those made by the
hammer of the smith or the saw of the carpenter, every one
knows how insensibly we throw these noises into a rhythmical
form in our own apprehension. We do this even without any
suggestion from the sounds themselves. For instance, if the
beats of a clock or watch be ever so exactly alike, we still
reckon them alternately tick-_tack_, tick-_tack_. That this
is the case, may be proved by taking a watch or clock of
such a construction that the returning swing of the pendulum
is silent, and in which therefore all the beats are
rigorously alike: we shall find ourselves still reckoning
its sounds as tick-_tack_. In this instance it is manifest
that the rhythm is entirely of our own making. In melodies,
also, and in verses in which the rhythm is complex, obscure
and difficult, we perceive something is required on our
part; for we are often incapable of contributing our share,
and thus lose the sense of the measure altogether. And when
we consider such cases, and attend to what passes within us
when we catch the measure, even of the simplest and
best-known air, we shall no longer doubt that an act of our
own thoughts is requisite in such cases, as well as
impressions on the sense. And thus the conception of this
peculiar modification of time, which we have called
_rhythm_, like all the other views which we have taken of
the subject, shows that we must, in order to form such
conceptions, supply a certain idea by our own thoughts, as
well as merely receive by senses, whether external or
internal, the impressions of appearances and collections of
appearances.
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NOTE TO CHAPTER X.
I HAVE in the last ten chapters described Space, Time, and
Number by various expressions, all intended to point out
their office as exemplifying the Ideal Element of human
knowledge. I have called them _Fundamental Ideas_; _Forms of
Perception_; _Forms of Intuition_; and perhaps other names.
I might add yet other phrases. I might say that the
properties of Space, Time, and Number are _Laws of the
Mind's Activity_ in apprehending what is. For the mind
cannot apprehend any thing or event except conformably to
the properties of space, time, and number. It is not only
that it _does_ not, but it _can_ not: and this impossibility
shows that the law is a law of the mind, and not of objects
extraneous to the mind.
It is usual for some of those who reject the doctrines here
presented to say that the axioms of geometry, and of other
sciences, are obtained by Induction from facts constantly
presented by experience. But I do not see how Induction can
prove that a proposition _must_ be true. The only
intelligible usage of the word _Induction_ appears to me to
be, that in which it is applied to a proposition which,
being separable from the facts in our apprehension, and
being compared with them, is seen to agree with them. But in
the cases now spoken of, the proposition is not separable
from the facts. We cannot infer by induction that two
straight lines cannot inclose a space, because we cannot
contemplate special cases of two lines inclosing a space, in
which it remains to be determined whether or not the
proposition, that both are straight, is true.
I do not deny that the activity of the mind by which it
perceives objects and events as related according to the
laws of space, time, and number, is awakened and developed
by being constantly exercised; and that we cannot imagine a
stage of human existence in which the powers have not been
awakened and {146} developed by such exercise. In this way,
experience and observation are necessary conditions and
prerequisites of our apprehension of geometrical (and other)
axioms. We cannot see the truth of these axioms without some
experience, because we cannot see any thing, or be human
beings, without some experience. This might be expressed by
saying that such truths are acquired necessarily _in the
course of_ all experience; but I think it is very
undesirable to apply, to such a case, the word _Induction_,
of which it is so important to us to keep the scientific
meaning free from confusion. Induction cannot give
demonstrative proofs, as I have already stated in Book 1. C.
i. sect. 3, and therefore cannot be the ground of necessary
truths.
Another expression which may be used to describe the
Fundamental Ideas here spoken of is suggested by the
language of a very profound and acute Review of the former
edition. The Reviewer holds that we pass from special
experiences to universal truths in virtue of 'the inductive
propensity--the irresistible impulse of the mind to
generalize _ad infinitum_.' I have already given reasons why
I cannot adopt the former expression; but I do not see why
space, time, number, cause, and the rest, may not be termed
_different forms_ of the _impulse of the mind to
generalize_. But if we put together all the Fundamental
Ideas as results of the Generalizing Impulse, we must still
separate them as different modes of action of that Impulse,
showing themselves in various characteristic ways in the
axioms and modes of reasoning which belong to different
sciences. The Generalizing Impulse in one case proceeds
according to the Idea of Space; in another, according to the
Idea of Mechanical Cause; and so in other subjects.
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