George Stuart Fullerton - An Introduction to Philosophy

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(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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