It does not seem absurd for a woman to Say: This piece of silk really
is yellow; it only looks white under this light. We all admit that a
white house may look pink under the rays of the setting sun, and we
never call it a pink house. We have seen that it is not unnatural to
say: That tree is really green; it is only its distance that makes it
look blue.
When one reflects upon these uses of the word "real," one recognizes
the fact that, among all the experiences in which things are revealed
to us, certain experiences impress us as being more prominent or
important or serviceable than certain others, and they come to be
called _real_. Things are not commonly seen by artificial light; the
sun is not always setting; the tree looks green when it is seen most
satisfactorily. In each case, the real color of the thing is the color
that it has under circumstances that strike us as normal or as
important. We cannot say that we always regard as most real that
aspect under which we most commonly perceive things, for if a more
unusual experience is more serviceable and really gives us more
information about the thing, we give the preference to that. Thus we
look with the naked eye at a moving speck on the table before us, and
we are unable to distinguish its parts. We place a microscope over the
speck and perceive an insect with all its members. The second
experience is the more unusual one, but would not every one say: Now we
perceive the thing _as it is_?
21. ULTIMATE REAL THINGS.--Let us turn away from the senses of the word
"real," which recognize one color or taste or odor as more real than
another, and come back to the real world of things presented in
sensations of touch. All other classes of sensations may be regarded
as related to this as the series of visual experiences above mentioned
was related to the one tree which was spoken of as revealed in them
all, the touch tree of which they gave information.
Can we say that this world is always to be regarded as reality and
never as appearance? We have already seen (section 8) that science
does not regard as anything more than appearance the real things which
seem to be directly presented in our experience.
This pen that I hold in my hand seems, as I pass my fingers over it, to
be continuously extended. It does not appear to present an alternation
of filled spaces and empty spaces. I am told that it is composed of
molecules in rapid motion and at considerable distances from one
another. I am further told that each molecule is composed of atoms,
and is, in its turn, not a continuous thing, but, so to speak, a group
of little things.
If I accept this doctrine, as it seems I must, am I not forced to
conclude that the reality which is given in my experience, the reality
with which I have contrasted appearances and to which I have referred
them, is, after all, itself only an appearance? The touch things which
I have hitherto regarded as the real things that make up the external
world, the touch things for which all my visual experiences have served
as signs, are, then, not themselves real external things, but only the
appearances under which real external things, themselves imperceptible,
manifest themselves to me.
It seems, then, that I do not directly perceive any real thing, or, at
least, anything that can be regarded as more than an appearance. What,
then, is the external world? What are things really like? Can we give
any true account of them, or are we forced to say with the skeptics
that we only know how things seem to us, and must abandon the attempt
to tell what they are really like?
Now, before one sets out to answer a question it is well to find out
whether it is a sensible question to ask and a sensible question to try
to answer. He who asks: Where is the middle of an infinite line? When
did all time begin? Where is space as a whole? does not deserve a
serious answer to his questions. And it is well to remember that he
who asks: What is the external world like? must keep his question a
significant one, if he is to retain his right to look for an answer at
all. He has manifestly no right to ask us: How does the external world
look when no one is looking? How do things feel when no one feels
them? How shall I think of things, not as I think of them, but as they
are?
If we are to give an account of the external world at all, it must
evidently be _an account_ of the external world; _i.e._ it must be
given in terms of our experience of things. The only legitimate
problem is to give a true account instead of a false one, to
distinguish between what only appears and is not real and what both
appears and is real.
Bearing this in mind, let us come back to the plain man's experience of
the world. He certainly seems to himself to perceive a real world of
things, and he constantly distinguishes, in a way very serviceable to
himself, between the merely apparent and the real. There is, of
course, a sense in which every experience is real; it is, at least, an
experience; but when he contrasts real and apparent he means something
more than this. Experiences are not relegated to this class or to that
merely at random, but the final decision is the outcome of a long
experience of the differences which characterize different individual
experiences and is an expression of the relations which are observed to
hold between them. Certain experiences are accepted as signs, and
certain others come to take the more dignified position of thing
signified; the mind rests in them and regards them as the real.
We have seen above that the world of real things in which the plain man
finds himself is a world of objects revealed in experiences of touch.
When he asks regarding anything: How far away is it? How big is it?
In what direction is it? it is always the touch thing that interests
him. What is given to the other senses is only a sign of this.
We have also seen (section 8) that the world of atoms and molecules of
which the man of science tells us is nothing more than a further
development of the world of the plain man. The real things with which
science concerns itself are, after all, only minute touch things,
conceived just as are the things with which the plain man is familiar.
They exist in space and move about in space, as the things about us are
perceived to exist in space and move about in space. They have size
and position, and are separated by distances. We do not _perceive_
them, it is true; but we _conceive_ them after the analogy of the
things that we do perceive, and it is not inconceivable that, if our
senses were vastly more acute, we might perceive them directly.
Now, when we conclude that the things directly perceptible to the sense
of touch are to be regarded as appearances, as signs of the presence of
these minuter things, do we draw such a conclusion arbitrarily? By no
means. The distinction between appearance and reality is drawn here
just as it is drawn in the world of our common everyday experiences.
The great majority of the touch things about us we are not actually
touching at any given moment. We only _see_ the things, _i.e._ we have
certain _signs_ of their presence. None the less we believe that the
things exist all the time. And in the same way the man of science does
not doubt the existence of the real things of which he speaks; he
perceives their _signs_. That certain experiences are to be taken as
signs of such realities he has established by innumerable observations
and careful deductions from those observations. To see the full force
of his reasonings one must read some work setting forth the history of
the atomic theory.
If, then, we ask the question: What is the real external world? it is
clear that we cannot answer it satisfactorily without taking into
consideration the somewhat shifting senses of the word "real." What is
the real external world to the plain man? It is the world of touch
things, of objects upon which he can lay his hands. What is the real
external world to the man of science? It is the world of atoms and
molecules, of minuter touch things that he cannot actually touch, but
which he conceives as though he could touch them.
It should be observed that the man of science has no right to deny the
real world which is revealed in the experience of the plain man. In
all his dealings with the things which interest him in common life, he
refers to this world just as the plain man does. He sees a tree and
walks towards it, and distinguishes between its real and its apparent
color, its real and its apparent size. He talks about seeing things as
they are, or not seeing things as they are. These distinctions in his
experience of things remain even after he has come to believe in atoms
and molecules.
Thus, the touch object, the tree as he feels it under his hand, may
come to be regarded as the sign of the presence of those entities that
science seems, at present, to regard as ultimate. Does this prevent it
from being the object which has stood as the interpreter of all those
diverse visual sensations that we have called different views of the
tree? They are still the appearances, and it, relatively to them, is
the reality. Now we find that it, in its turn, can be used as a sign
of something else, can be regarded as an appearance of a reality more
ultimate. It is clear, then, that the same thing may be regarded both
as appearance and as reality--appearance as contrasted with one thing,
and reality as contrasted with another.
But suppose one says: _I do not want to know what the real external
world is to this man or to that man; I want to know what the real
external world is_. What shall we say to such a demand?
There is a sense in which such a demand is not purely meaningless,
though it may not be a very sensible demand to make. We have seen that
an increase of knowledge about things compels a man to pass from the
real things of common life to the real things of science, and to look
upon the former as appearance. Now, a man may arbitrarily decide that
he will use the word "reality" to indicate only that which can never in
its turn be regarded as appearance, a reality which must remain an
ultimate reality; and he may insist upon our telling him about that.
How a man not a soothsayer can tell when he has come to ultimate
reality, it is not easy to see.
Suppose, however, that we could give any one such information. We
should then be telling him about things _as they are_, it is true, but
his knowledge of things would not be different in _kind_ from what it
was before. The only difference between such a knowledge of things and
a knowledge of things not known to be ultimate would be that, in the
former case, it would be recognized that no further extension of
knowledge was possible. The distinction between appearance and reality
would remain just what it was in the experience of the plain man.
22. THE BUGBEAR OF THE "UNKNOWABLE."--It is very important to recognize
that we must not go on talking about appearance and reality, as if our
words really meant something, when we have quite turned our backs upon
our experience of appearances and the realities which they represent.
That appearances and realities are connected we know very well, for we
perceive them to be connected. What we see, we can touch. And we not
only know that appearances and realities are connected, but we know
with much detail what appearances are to be taken as signs of what
realities. The visual experience which I call the house as seen from a
distance I never think of taking for a representative of the hat which
I hold in my hand. This visual experience I refer to its own
appropriate touch thing, and not to another. If what _looks like_ a
beefsteak could _really be_ a fork or a mountain or a kitten
indifferently,--but I must not even finish the sentence, for the words
"look like" and "could really be" lose all significance when we loosen
the bond between appearances and the realities to which they are
properly referred.
Each appearance, then, must be referred to some particular real thing
and not to any other. This is true of the appearances which we
recognize as such in common life, and it is equally true of the
appearances recognized as such in science. The pen which I feel
between my fingers I may regard as appearance and refer to a swarm of
moving atoms. But it would be silly for me to refer it to atoms "in
general." The reality to which I refer the appearance in question is a
particular group of atoms existing at a particular point in space. The
chemist never supposes that the atoms within the walls of his test-tube
are identical with those in the vial on the shelf. Neither in common
life nor in science would the distinction between appearances and real
things be of the smallest service were it not possible to distinguish
between this appearance and that, and this reality and that, and to
refer each appearance to its appropriate reality. Indeed, it is
inconceivable that, under such circumstances, the distinction should
have been drawn at all.
These points ought to be strongly insisted upon, for we find certain
philosophic writers falling constantly into a very curious abuse of the
distinction and making much capital of it. It is argued that what we
see, what we touch, what we conceive as a result of scientific
observation and reflection--all is, in the last analysis, material
which is given us in sensation. The various senses furnish us with
different classes of sensations; we work these up into certain
complexes. But sensations are only the impressions which something
outside of us makes upon us. Hence, although we seem to ourselves to
know the external world as it is, our knowledge can never extend beyond
the impressions made upon us. Thus, we are absolutely shut up to
_appearances_, and can know nothing about the _reality_ to which they
must be referred.
Touching this matter Herbert Spencer writes[1] as follows: "When we are
taught that a piece of matter, regarded by us as existing externally,
cannot be really known, but that we can know only certain impressions
produced on us, we are yet, by the relativity of thought, compelled to
think of these in relation to a cause--the notion of a real existence
which generated these impressions becomes nascent. If it be proved
that every notion of a real existence which we can frame is
inconsistent with itself,--that matter, however conceived by us, cannot
be matter as it actually is,--our conception, though transfigured, is
not destroyed: there remains the sense of reality, dissociated as far
as possible from those special forms under which it was before
represented in thought."
This means, in plain language, that we must regard everything we know
and can know as appearance and must refer it to an unknown reality.
Sometimes Mr. Spencer calls this reality the Unknowable, sometimes he
calls it the Absolute, and sometimes he allows it to pass by a variety
of other names, such as Power, Cause, etc. He wishes us to think of it
as "lying behind appearances" or as "underlying appearances."
Probably it has already been remarked that this Unknowable has brought
us around again to that amusing "telephone exchange" discussed in the
third chapter. But if the reader feels within himself the least
weakness for the Unknowable, I beg him to consider carefully, before he
pins his faith to it, the following:--
(1) If we do perceive external bodies, our own bodies and others, then
it is conceivable that we may have evidence from observation to the
effect that other bodies affecting our bodies may give rise to
sensations. In this case we cannot say that we know nothing but
sensations; we know real bodies as well as sensations, and we may refer
the sensations to the real bodies.
(2) If we do not perceive that we have bodies, and that our bodies are
acted upon by others, we have no evidence that what we call our
sensations are due to messages which come from "external things" and
are conducted along the nerves. It is then, absurd to talk of such
"external things" as though they existed, and to call them the reality
to which sensations, as appearances, must be referred,
(3) In other words, if there is perceived to be a telephone exchange
with its wires and subscribers, we may refer the messages received to
the subscribers, and call this, if we choose, a reference of appearance
to reality.
But if there is perceived no telephone exchange, and if it is concluded
that any wires or subscribers of which it means anything to speak must
be composed of what we have heretofore called "messages," then it is
palpably absurd to refer the "messages" as a whole to subscribers not
supposed to be composed of "messages"; and it is a blunder to go on
calling the things that we know "messages," as though we had evidence
that they came from, and must be referred to, something beyond
themselves.
We must recognize that, with the general demolition of the exchange, we
lose not only known subscribers, but the very notion of a subscriber.
It will not do to try to save from this wreck some "unknowable"
subscriber, and still pin our faith to him.
(4) We have seen that the relation of appearance to reality is that of
certain experiences to certain other experiences. When we take the
liberty of calling the Unknowable a _reality_, we blunder in our use of
the word. The Unknowable cannot be an experience either actual,
possible, or conceived as possible, and it cannot possibly hold the
relation to any of our experiences that a real thing of any kind holds
to the appearances that stand as its signs.
(5) Finally, no man has ever made an assumption more perfectly useless
and purposeless than the assumption of the Unknowable. We have seen
that the distinction between appearance and reality is a serviceable
one, and it has been pointed out that it would be of no service
whatever if it were not possible to refer particular appearances to
their own appropriate realities. The realities to which we actually
refer appearances serve to explain them. Thus, when I ask: Why do I
perceive that tree now as faint and blue and now as vivid and green?
the answer to the question is found in the notion of distance and
position in space; it is found, in other words, in a reference to the
real world of touch things, for which visual experiences serve as
signs. Under certain circumstances, the mountain _ought_ to be robed
in its azure hue, and, under certain circumstances, it _ought not_.
The circumstances in each case are open to investigation.
Now, let us substitute for the real world of touch things, which
furnishes the explanation of given visual experiences, that philosophic
fiction, that pseudo-real nonentity, the Unknowable. Now I perceive a
tree as faint and blue, now as bright and green; will a reference to
the Unknowable explain why the experiences differed? Was the
Unknowable in the one instance farther off in an unknowable space, and
in the other nearer? This, even if it means anything, must remain
unknowable. And when the chemist puts together a volume of chlorine
gas and a volume of hydrogen gas to get two volumes of hydrochloric
acid gas, shall we explain the change which has taken place by a
reference to the Unknowable, or shall we turn to the doctrine of atoms
and their combinations?
The fact is that no man in his senses tries to account for any
individual fact by turning for an explanation to the Unknowable. It is
a life-preserver by which some set great store, but which no man dreams
of using when he really falls into the water.
If, then, we have any reason to believe that there is a real external
world at all, we have reason to believe that we know what it is. That
some know it imperfectly, that others know it better, and that we may
hope that some day it will be known still more perfectly, is surely no
good reason for concluding that we do not know it at all.
[1] "First Principles," Part I, Chapter IV, section 26.
CHAPTER VI
OF SPACE
23. WHAT ARE WE SUPPOSED TO KNOW ABOUT IT.--The plain man may admit
that he is not ready to hazard a definition of space, but he is
certainly not willing to admit that he is wholly ignorant of space and
of its attributes. He knows that it is something in which material
objects have position and in which they move about; he knows that it
has not merely length, like a line, nor length and breadth, like a
surface, but has the three dimensions of length, breadth, and depth; he
knows that, except in the one circumstance of its position, every part
of space is exactly like every other part, and that, although objects
may move about in space, it is incredible that the spaces themselves
should be shifted about.
Those who are familiar with the literature of the subject know that it
has long been customary to make regarding space certain other
statements to which the plain man does not usually make serious
objection when he is introduced to them. Thus it is said:--
(1) The idea of space is _necessary_. We can think of objects in space
as annihilated, but we cannot conceive space to be annihilated. We can
clear space of things, but we cannot clear away space itself, even in
thought.
(2) Space must be _infinite_. We cannot conceive that we should come
to the end of space.
(3) Every space, however small, is _infinitely divisible_. That is to
say, even the most minute space must be composed of spaces. We cannot,
even theoretically, split a solid into mere surfaces, a surface into
mere lines, or a line into mere points.
Against such statements the plain man is not impelled to rise in
rebellion, for he can see that there seems to be some ground for making
them. He can conceive of any particular material object as
annihilated, and of the place which it occupied as standing empty; but
he cannot go on and conceive of the annihilation of this bit of empty
space. Its annihilation would not leave a gap, for a gap means a bit
of empty space; nor could it bring the surrounding spaces into
juxtaposition, for one cannot shift spaces, and, in any case, a
shifting that is not a shifting through space is an absurdity.
Again, he cannot conceive of any journey that would bring him to the
end of space. There is no more reason for stopping at one point than
at another; why not go on? What could end space?
As to the infinite divisibility of space, have we not, in addition to
the seeming reasonableness of the doctrine, the testimony of all the
mathematicians? Does any one of them ever dream of a line so short
that it cannot be divided into two shorter lines, or of an angle so
small that it cannot be bisected?
24. SPACE AS NECESSARY AND SPACE AS INFINITE.--That these statements
about space contain truth one should not be in haste to deny. It seems
silly to say that space can be annihilated, or that one can travel
"over the mountains of the moon" in the hope of reaching the end of it.
And certainly no prudent man wishes to quarrel with that coldly
rational creature the mathematician.
But it is well worth while to examine the statements carefully and to
see whether there is not some danger that they may be understood in
such a way as to lead to error. Let us begin with the doctrine that
space is necessary and cannot be "thought away."
As we have seen above, it is manifestly impossible to annihilate in
thought a certain portion of space and leave the other portions intact.
There are many things in the same case. We cannot annihilate in
thought one side of a door and leave the other side; we cannot rob a
man of the outside of his hat and leave him the inside. But we can
conceive of a whole door as annihilated, and of a man as losing a whole
hat. May we or may we not conceive of space as a whole as nonexistent?
I do not say, be it observed, can we conceive of something as attacking
and annihilating space? Whatever space may be, we none of us think of
it as a something that may be threatened and demolished. I only say,
may we not think of a system of things--not a world such as ours, of
course, but still a system of things of some sort--in which space
relations have no part? May we not conceive such to be possible?
It should be remarked that space relations are by no means the only
ones in which we think of things as existing. We attribute to them
time relations as well. Now, when we think of occurrences as related
to each other in time, we do, in so far as we concentrate our attention
upon these relations, turn our attention away from space and
contemplate another aspect of the system of things. Space is not such
a necessity of thought that we must keep thinking of space when we have
turned our attention to something else. And is it, indeed,
inconceivable that there should be a system of things (not extended
things in space, of course), characterized by time relations and
perhaps other relations, but not by space relations?
It goes without saying that we cannot go on thinking of space and at
the same time not think of space. Those who keep insisting upon space
as a necessity of thought seem to set us such a task as this, and to
found their conclusion upon our failure to accomplish it. "We can
never represent to ourselves the nonexistence of space," says the
German philosopher Kant (1724-1804), "although we can easily conceive
that there are no objects in space."
It would, perhaps, be fairer to translate the first half of this
sentence as follows: "We can never picture to ourselves the
nonexistence of space." Kant says we cannot make of it a
_Vorstellung_, a representation. This we may freely admit, for what
does one try to do when one makes the effort to imagine the
nonexistence of space? Does not one first clear space of objects, and
then try to clear space of space in much the same way? We try to
"think space away," _i.e. to remove it from the place where it was and
yet keep that place_.
What does it mean to imagine or represent to oneself the nonexistence
of material objects? Is it not to represent to oneself the objects as
no longer in space, _i.e._ to imagine the space as empty, as cleared of
the objects? It means something in this case to speak of a
_Vorstellung_, or representation. We can call before our minds the
empty space. But if we are to think of space as nonexistent, what
shall we call before our minds? Our procedure must not be analogous to
what it was before; we must not try to picture to our minds _the
absence of space_, as though that were in itself a something that could
be pictured; we must turn our attention to other relations, such as
time relations, and ask whether it is not conceivable that such should
be the only relations obtaining within a given system.
Those who insist upon the fact that we cannot but conceive space as
infinite employ a very similar argument to prove their point. They set
us a self-contradictory task, and regard our failure to accomplish it
as proof of their position. Thus, Sir William Hamilton (1788-1856)
argues: "We are altogether unable to conceive space as bounded--as
finite; that is, as a whole beyond which there is no further space."
And Herbert Spencer echoes approvingly: "We find ourselves totally
unable to imagine bounds beyond which there is no space."
Now, whatever one may be inclined to think about the infinity of space,
it is clear that this argument is an absurd one. Let me write it out
more at length: "We are altogether unable to conceive space as
bounded--as finite; that is, as a whole _in the space_ beyond which
there is no further space." "We find ourselves totally unable to
imagine bounds, _in the space_ beyond which there is no further space."
The words which I have added were already present implicitly. What can
the word "beyond" mean if it does not signify space beyond? What Sir
William and Mr. Spencer have asked us to do is to imagine a limited
space with a _beyond_ and yet _no beyond_.
There is undoubtedly some reason why men are so ready to affirm that
space is infinite, even while they admit that they do not know that the
world of material things is infinite. To this we shall come back again
later. But if one wishes to affirm it, it is better to do so without
giving a reason than it is to present such arguments as the above.
25. SPACE AS INFINITELY DIVISIBLE.--For more than two thousand years
men have been aware that certain very grave difficulties seem to attach
to the idea of motion, when we once admit that space is infinitely
divisible. To maintain that we can divide any portion of space up into
ultimate elements which are not themselves spaces, and which have no
extension, seems repugnant to the idea we all have of space. And if we
refuse to admit this possibility there seems to be nothing left to us
but to hold that every space, however small, may theoretically be
divided up into smaller spaces, and that there is no limit whatever to
the possible subdivision of spaces. Nevertheless, if we take this most
natural position, we appear to find ourselves plunged into the most
hopeless of labyrinths, every turn of which brings us face to face with
a flat self-contradiction.
To bring the difficulties referred to clearly before our minds, let us
suppose a point to move uniformly over a line an inch long, and to
accomplish its journey in a second. At first glance, there appears to
be nothing abnormal about this proceeding. But if we admit that this
line is infinitely divisible, and reflect upon this property of the
line, the ground seems to sink from beneath our feet at once.
For it is possible to argue that, under the conditions given, the point
must move over one half of the line in half a second; over one half of
the remainder, or one fourth of the line, in one fourth of a second;
over one eighth of the line, in one eighth of a second, etc. Thus the
portions of line moved over successively by the point may be
represented by the descending series:
1/2, 1/4, 1/8, 1/16, . . . [Greek omicron symbol]
Now, it is quite true that the motion of the point can be described in
a number of different ways; but the important thing to remark here is
that, if the motion really is uniform, and if the line really is
infinitely divisible, this series must, as satisfactorily as any other,
describe the motion of the point. And it would be absurd to maintain
that _a part_ of the series can describe the whole motion. We cannot
say, for example, that, when the point has moved over one half, one
fourth, and one eighth of the line, it has completed its motion. If
even a single member of the series is left out, the whole line has not
been passed over; and this is equally true whether the omitted member
represent a large bit of line or a small one.
The whole series, then, represents the whole line, as definite parts of
the series represent definite parts of the line. The line can only be
completed when the series is completed. But when and how can this
series be completed? In general, a series is completed when we reach
the final term, but here there appears to be no final term. We cannot
make zero the final term, for it does not belong to the series at all.
It does not obey the law of the series, for it is not one half as large
as the term preceding it--what space is so small that dividing it by 2
gives us [omicron]? On the other hand, some term just before zero
cannot be the final term; for if it really represents a little bit of
the line, however small, it must, by hypothesis, be made up of lesser
bits, and a smaller term must be conceivable. There can, then, be no
last term to the series; _i.e._ what the point is doing at the very
last is absolutely indescribable; it is inconceivable that there should
be a _very last_.
It was pointed out many centuries ago that it is equally inconceivable
that there should be a _very first_. How can a point even begin to
move along an infinitely divisible line? Must it not before it can
move over any distance, however short, first move over half that
distance? And before it can move over that half, must it not move over
the half of that? Can it find something to move over that has no
halves? And if not, how shall it even start to move? To move at all,
it must begin somewhere; it cannot begin with what has no halves, for
then it is not moving over any part of the line, as all parts have
halves; and it cannot begin with what has halves, for that is not the
beginning. _What does the point do first?_ that is the question.
Those who tell us about points and lines usually leave us to call upon
gentle echo for an answer.
The perplexities of this moving point seem to grow worse and worse the
longer one reflects upon them. They do not harass it merely at the
beginning and at the end of its journey. This is admirably brought out
by Professor W. K. Clifford (1845-1879), an excellent mathematician,
who never had the faintest intention of denying the possibility of
motion, and who did not desire to magnify the perplexities in the path
of a moving point. He writes:--
"When a point moves along a line, we know that between any two
positions of it there is an infinite number . . . of intermediate
positions. That is because the motion is continuous. Each of those
positions is where the point was at some instant or other. Between the
two end positions on the line, the point where the motion began and the
point where it stopped, there is no point of the line which does not
belong to that series. We have thus an infinite series of successive
positions of a continuously moving point, and in that series are
included all the points of a certain piece of line-room." [1]
Thus, we are told that, when a point moves along a line, between any
two positions of it there is an infinite number of intermediate
positions. Clifford does not play with the word "infinite"; he takes
it seriously and tells us that it means without any end: "_Infinite_;
it is a dreadful word, I know, until you find out that you are familiar
with the thing which it expresses. In this place it means that between
any two positions there is some intermediate position; between that and
either of the others, again, there is some other intermediate; and so
on _without any end_. Infinite means without any end."
But really, if the case is as stated, the point in question must be at
a desperate pass. I beg the reader to consider the following, and ask
himself whether he would like to change places with it:--
(1) If the series of positions is really endless, the point must
complete one by one the members of an endless series, and reach a
nonexistent final term, for a really endless series cannot have a final
term.
(2) The series of positions is supposed to be "an infinite series of
successive positions." The moving point must take them one after
another. But how can it? _Between any two positions of the point
there is an infinite number of intermediate positions_. That is to
say, no two of these successive positions must be regarded as _next to_
each other; every position is separated from every other by an infinite
number of intermediate ones. How, then, shall the point move? It
cannot possibly move from one position to the next, for there is no
next. Shall it move first to some position that is not the next? Or
shall it in despair refuse to move at all?
Evidently there is either something wrong with this doctrine of the
infinite divisibility of space, or there is something wrong with our
understanding of it, if such absurdities as these refuse to be cleared
away. Let us see where the trouble lies.
26. WHAT IS REAL SPACE?--It is plain that men are willing to make a
number of statements about space, the ground for making which is not at
once apparent. It is a bold man who will undertake to say that the
universe of matter is infinite in extent. We feel that we have the
right to ask him how he knows that it is. But most men are ready
enough to affirm that space is and must be infinite. How do they know
that it is? They certainly do not directly perceive all space, and
such arguments as the one offered by Hamilton and Spencer are easily
seen to be poor proofs.
Men are equally ready to affirm that space is infinitely divisible.
Has any man ever looked upon a line and perceived directly that it has
an infinite number of parts? Did any one ever succeed in dividing a
space up infinitely? When we try to make clear to ourselves how a
point moves along an infinitely divisible line, do we not seem to land
in sheer absurdities? On what sort of evidence does a man base his
statements regarding space? They are certainly very bold statements.
A careful reflection reveals the fact that men do not speak as they do
about space for no reason at all. When they are properly understood,
their statements can be seen to be justified, and it can be seen also
that the difficulties which we have been considering can be avoided.
The subject is a deep one, and it can scarcely be discussed
exhaustively in an introductory volume of this sort, but one can, at
least, indicate the direction in which it seems most reasonable to look
for an answer to the questions which have been raised. How do we come
to a knowledge of space, and what do we mean by space? This is the
problem to solve; and if we can solve this, we have the key which will
unlock many doors.
Now, we saw in the last chapter that we have reason to believe that we
know what the real external world is. It is a world of things which we
perceive, or can perceive, or, not arbitrarily but as a result of
careful observation and deductions therefrom, conceive as though we did
perceive it--a world, say, of atoms and molecules. It is not an
Unknowable behind or beyond everything that we perceive, or can
perceive, or conceive in the manner stated.
And the space with which we are concerned is real space, the space in
which real things exist and move about, the real things which we can
directly know or of which we can definitely know something. In some
sense it must be given in our experience, if the things which are in
it, and are known to be in it, are given in our experience. How must
we think of this real space?
Suppose we look at a tree at a distance. We are conscious of a certain
complex of color. We can distinguish the kind of color; in this case,
we call it blue. But the quality of the color is not the only thing
that we can distinguish in the experience. In two experiences of color
the quality may be the same, and yet the experiences may be different
from each other. In the one case we may have more of the same
color--we may, so to speak, be conscious of a larger patch; but even if
there is not actually more of it, there may be such a difference that
we can know from the visual experience alone that the touch object
before us is, in the one case, of the one shape, and, in the other
case, of another. Thus we may distinguish between the _stuff_ given in
our experience and the _arrangement_ of that stuff. This is the
distinction which philosophers have marked as that between "matter" and
"form." It is, of course, understood that both of these words, so
used, have a special sense not to be confounded with their usual one.
This distinction between "matter" and "form" obtains in all our
experiences. I have spoken just above of the shape of the touch object
for which our visual experiences stand as signs. What do we mean by
its shape? To the plain man real things are the touch things of which
he has experience, and these touch things are very clearly
distinguishable from one another in shape, in size, in position, nor
are the different parts| of the things to be confounded with each
other. Suppose that, as we pass our hand over a table, all the
sensations of touch and movement which we experience fused into an
undistinguishable mass. Would we have any notion of size or shape? It
is because our experiences of touch and movement do not fuse, but
remain distinguishable from each other, and we are conscious of them as
_arranged_, as constituting a system, that we can distinguish between
this part of a thing and that, this thing and that.
This arrangement, this order, of what is revealed by touch and
movement, we may call the "form" of the touch world. Leaving out of
consideration, for the present, time relations, we may say that the
"form" of the touch world is the whole system of actual and possible
relations of arrangement between the elements which make it up. It is
because there is such a system of relations that we can speak of things
as of this shape or of that, as great or small, as near or far, as here
or there.
Now, I ask, is there any reason to believe that, when the plain man
speaks of _space_, the word means to him anything more than this system
of actual and possible relations of arrangement among the touch things
that constitute his real world? He may talk sometimes as though space
were some kind of a _thing_, but he does not really think of it as a
thing.
This is evident from the mere fact that he is so ready to make about it
affirmations that he would not venture to make about things. It does
not strike him as inconceivable that a given material object should be
annihilated; it does strike him as inconceivable that a portion of
space should be blotted out of existence. Why this difference? Is it
not explained when we recognize that space is but a name for all the
actual and possible relations of arrangement in which things in the
touch world may stand? We cannot drop out some of these relations and
yet keep _space_, _i.e._ the system of relations which we had before.
That this is what space means, the plain man may not recognize
explicitly, but he certainly seems to recognize it implicitly in what
he says about space. Men are rarely inclined to admit that space is a
_thing_ of any kind, nor are they much more inclined to regard it as a
quality of a thing. Of what could it be the quality?
And if space really were a thing of any sort, would it not be the
height of presumption for a man, in the absence of any direct evidence
from observation, to say how much there is of it--to declare it
infinite? Men do not hesitate to say that space must be infinite. But
when we realize that we do not mean by space merely the actual
relations which exist between the touch things that make up the world,
but also the _possible_ relations, _i.e._ that we mean the whole _plan_
of the world system, we can see that it is not unreasonable to speak of
space as infinite.
The material universe may, for aught we know, be limited in extent.
The actual space relations in which things stand to each other may not
be limitless. But these actual space relations taken alone do not
constitute space. Men have often asked themselves whether they should
conceive of the universe as limited and surrounded by void space. It
is not nonsense to speak of such a state of things. It would, indeed,
appear to be nonsense to say that, if the universe is limited, it does
not lie in void space. What can we mean by void space but the system
of possible relations in which things, if they exist, must stand? To
say that, beyond a certain point, no further relations are possible,
seems absurd.
Hence, when a man has come to understand what we have a right to mean
by space, it does not imply a boundless conceit on his part to hazard
the statement that space is infinite. When he has said this, he has
said very little. What shall we say to the statement that space is
infinitely divisible?
To understand the significance of this statement we must come back to
the distinction between appearances and the real things for which they
stand as signs, the distinction discussed at length in the last chapter.
When I see a tree from a distance, the visual experience which I have
is, as we have seen, not an indivisible unit, but is a complex
experience; it has parts, and these parts are related to each other; in
other words, it has both "matter" and "form." It is, however, one
thing to say that this experience has parts, and it is another to say
that it has an infinite number of parts. No man is conscious of
perceiving an infinite number of parts in the patch of color which
represents to him a tree at a distance; to say that it is constituted
of such strikes us in our moments of sober reflection as a monstrous
statement.
Now, this visual experience is to us the sign of the reality, the real
tree; it is not taken as the tree itself. When we speak of the size,
the shape, the number of parts, of the tree, we do not have in mind the
size, the shape, the number of parts, of just this experience. We pass
from the sign to the thing signified, and we may lay our hand upon this
thing, thus gaining a direct experience of the size and shape of the
touch object.
We must recognize, however, that just as no man is conscious of an
infinite number of parts in what he sees, so no man is conscious of an
infinite number of parts in what he touches. He who tells me that,
when I pass my finger along my paper cutter, _what I perceive_ has an
infinite number of parts, tells me what seems palpably untrue. When an
object is very small, I can see it, and I cannot see that it is
composed of parts; similarly, when an object is very small, I can feel
it with my finger, but I cannot distinguish its parts by the sense of
touch. There seem to be limits beyond which I cannot go in either case.
Nevertheless, men often speak of thousandths of an inch, or of
millionths of an inch, or of distances even shorter. Have such
fractions of the magnitudes that we do know and can perceive any real
existence? The touch world of real things as it is revealed in our
experience does not appear to be divisible into such; it does not
appear to be divisible even so far, and much less does it appear to be
infinitely divisible.
But have we not seen that the touch world given in our experience must
be taken by the thoughtful man as itself the sign or appearance of a
reality more ultimate? The speck which appears to the naked eye to
have no parts is seen under the microscope to have parts; that is to
say, an experience apparently not extended has become the sign of
something that is seen to have part out of part. We have as yet
invented no instrument that will make directly perceptible to the
finger tip an atom of hydrogen or of oxygen, but the man of science
conceives of these little things as though they could be perceived.
They and the space in which they move--the system of actual and
possible relations between them--seem to be related to the world
revealed in touch very much as the space revealed in the field of the
microscope is related to the space of the speck looked at with the naked eye. |
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