The Project Gutenberg eBook, An Introduction to Philosophy, by George

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 
 
53

---

 Chap. VI – Of Space 
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 not 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 
 
54

---

 Chap. VI – Of Space 
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