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Page 9
The usual answer to this question is "No," but that is not the right
answer. The purpose of the following remarks is to show that the
answer should be "Yes." I want to show that without any extraordinary
difficulty we can illustrate the theory of a finite universe by
means of a mental image to which, with some practice, we shall soon
grow accustomed.
First of all, an observation of epistemological nature. A
geometrical-physical theory as such is incapable of being directly
pictured, being merely a system of concepts. But these concepts
serve the purpose of bringing a multiplicity of real or imaginary
sensory experiences into connection in the mind. To "visualise"
a theory, or bring it home to one's mind, therefore means to give
a representation to that abundance of experiences for which the
theory supplies the schematic arrangement. In the present case we
have to ask ourselves how we can represent that relation of solid
bodies with respect to their reciprocal disposition (contact) which
corresponds to the theory of a finite universe. There is really
nothing new in what I have to say about this; but innumerable
questions addressed to me prove that the requirements of those who
thirst for knowledge of these matters have not yet been completely
satisfied.
So, will the initiated please pardon me, if part of what I shall
bring forward has long been known?
What do we wish to express when we say that our space is infinite?
Nothing more than that we might lay any number whatever of bodies
of equal sizes side by side without ever filling space. Suppose
that we are provided with a great many wooden cubes all of the
same size. In accordance with Euclidean geometry we can place them
above, beside, and behind one another so as to fill a part of space
of any dimensions; but this construction would never be finished;
we could go on adding more and more cubes without ever finding
that there was no more room. That is what we wish to express when
we say that space is infinite. It would be better to say that space
is infinite in relation to practically-rigid bodies, assuming that
the laws of disposition for these bodies are given by Euclidean
geometry.
Another example of an infinite continuum is the plane. On a plane
surface we may lay squares of cardboard so that each side of any
square has the side of another square adjacent to it. The construction
is never finished; we can always go on laying squares--if their laws
of disposition correspond to those of plane figures of Euclidean
geometry. The plane is therefore infinite in relation to the
cardboard squares. Accordingly we say that the plane is an infinite
continuum of two dimensions, and space an infinite continuum of
three dimensions. What is here meant by the number of dimensions,
I think I may assume to be known.
Now we take an example of a two-dimensional continuum which is
finite, but unbounded. We imagine the surface of a large globe and
a quantity of small paper discs, all of the same size. We place
one of the discs anywhere on the surface of the globe. If we move
the disc about, anywhere we like, on the surface of the globe,
we do not come upon a limit or boundary anywhere on the journey.
Therefore we say that the spherical surface of the globe is an
unbounded continuum. Moreover, the spherical surface is a finite
continuum. For if we stick the paper discs on the globe, so that
no disc overlaps another, the surface of the globe will finally
become so full that there is no room for another disc. This simply
means that the spherical surface of the globe is finite in relation
to the paper discs. Further, the spherical surface is a non-Euclidean
continuum of two dimensions, that is to say, the laws of disposition
for the rigid figures lying in it do not agree with those of the
Euclidean plane. This can be shown in the following way. Place
a paper disc on the spherical surface, and around it in a circle
place six more discs, each of which is to be surrounded in turn
by six discs, and so on. If this construction is made on a plane
surface, we have an uninterrupted disposition in which there are
six discs touching every disc except those which lie on the outside.
[Figure 1: Discs maximally packed on a plane]
On the spherical surface the construction also seems to promise
success at the outset, and the smaller the radius of the disc
in proportion to that of the sphere, the more promising it seems.
But as the construction progresses it becomes more and more patent
that the disposition of the discs in the manner indicated, without
interruption, is not possible, as it should be possible by Euclidean
geometry of the the plane surface. In this way creatures which
cannot leave the spherical surface, and cannot even peep out from
the spherical surface into three-dimensional space, might discover,
merely by experimenting with discs, that their two-dimensional
"space" is not Euclidean, but spherical space.
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