An Introduction to Philosophy

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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;

customary to make regarding space certain other statements to which the plain man doe

annihilated, but we cannot conceive space to be annihilated. We can clea

e cannot conceive that we sho

most minute space must be composed of spaces. We cannot, even theoretically, split

nihilated, 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 me

to the end of space. There is no more reason for stopping at o

e doctrine, the testimony of all the mathematicians? Does any one of them ever dream of a line so shor

ny. It seems silly to say that space can be annihilated, or that one can travel "over the mountains of the moon" in the hope o

ere is not some danger that they may be understood in such a way as to lead to error

s 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

nk 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 ou

ar 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 tur

ssity 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 our

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 clea

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 t

d 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 i

inite; 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

they do not know that the world of material things is infinite. To this we shall come back again later. But if one

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 smal

to accomplish its journey in a second. At first glance, there appears to be nothing abnormal about this proceeding. But if we admit tha

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 eigh

/16, . . . [Gree

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,

l 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

d 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 mo

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

ions 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 doe

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

be at a desperate pass. I beg the reader to consider the followin

lete one by one the members of an endless series, and reach a nonexis

te 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 one

ibility of space, or there is something wrong with our understanding of it, if suc

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

f 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 lin

ch 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 reasonabl

, 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,

s which we can directly know or of which we can definitely know something. In some sense it must be given in our experience,

ifferent 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 o

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 m

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 i

ore than this system of actual and possible relations of arrangement among the touch things that constitute his real

ence. 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

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

en 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 li

a boundless conceit on his part to hazard the statement that space is infinite. When he has said t

he distinction between appearances and the real things for which they s

her 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 perceivin

hape, 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 th

hen 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 comp

he 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 n

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

to his sense of touch. He may speak of real things too small to be thus perceived, and of their motion as through spaces too small to be perceptible at all. What limit shall he set to the poss

that we may carry on our division in the case of the latter, when a further subdivision of the former seems out of the question. But it should not mean

The line as revealed in a single experience either of sight or of touch is not composed of an infinite number of parts. It is composed of points se

e may be. This only means that the visual or tactual point of the single experience may stand for, may represent, what is not a mere point but has part

when he tells us that between any two points on a line there are an infinite number of other points, he only means that we may expand the line indefinitely by the system of substitutions described above. We do this for ourselves within limits ev

Thinking," p. 14