THE SIGNIFICANCE OF A FOUR-DIMENSIONAL EXISTENCE
Having now obtained the conception of a four-dimensional space, and
having formed the analogy which, without any further geometrical
difficulties, enables us to enquire into its properties, I will refer
the reader, whose interest is principally in the mechanical aspect,
to Chapters VI. and VII. In the present chapter I will deal with
the general significance of the enquiry, and in the next with the
historical origin of the idea.
First, with regard to the question of whether there is any evidence
that we are really in four-dimensional space, I will go back to the
analogy of the plane world.
A being in a plane world could not have any experience of
three-dimensional shapes, but he could have an experience of
three-dimensional movements.
We have seen that his matter must be supposed to have an extension,
though a very small one, in the third dimension. And thus, in the
small particles of his matter, three-dimensional movements may well
be conceived to take place. Of these movements he would only perceive
the resultants. Since all movements of an observable size in the plane
world are two-dimensional, he would only perceive the resultants in
two dimensions of the small three-dimensional movements. Thus, there
would be phenomena which he could not explain by his theory of
mechanics—motions would take place which he could not explain by his
theory of motion. Hence, to determine if we are in a four-dimensional
world, we must examine the phenomena of motion in our space. If
movements occur which are not explicable on the suppositions of our
three-dimensional mechanics, we should have an indication of a possible
four-dimensional motion, and if, moreover, it could be shown that such
movements would be a consequence of a four-dimensional motion in the
minute particles of bodies or of the ether, we should have a strong
presumption in favour of the reality of the fourth dimension.
By proceeding in the direction of finer and finer subdivision, we come
to forms of matter possessing properties different to those of the
larger masses. It is probable that at some stage in this process we
should come to a form of matter of such minute subdivision that its
particles possess a freedom of movement in four dimensions. This form
of matter I speak of as four-dimensional ether, and attribute to it
properties approximating to those of a perfect liquid.
Deferring the detailed discussion of this form of matter to Chapter
VI., we will now examine the means by which a plane being would come to
the conclusion that three-dimensional movements existed in his world,
and point out the analogy by which we can conclude the existence of
four-dimensional movements in our world. Since the dimensions of the
matter in his world are small in the third direction, the phenomena in
which he would detect the motion would be those of the small particles
of matter.
Suppose that there is a ring in his plane. We can imagine currents
flowing round the ring in either of two opposite directions. These
would produce unlike effects, and give rise to two different fields
of influence. If the ring with a current in it in one direction be
taken up and turned over, and put down again on the plane, it would be
identical with the ring having a current in the opposite direction. An
operation of this kind would be impossible to the plane being. Hence
he would have in his space two irreconcilable objects, namely, the
two fields of influence due to the two rings with currents in them in
opposite directions. By irreconcilable objects in the plane I mean
objects which cannot be thought of as transformed one into the other by
any movement in the plane.
Instead of currents flowing in the rings we can imagine a different
kind of current. Imagine a number of small rings strung on the original
ring. A current round these secondary rings would give two varieties
of effect, or two different fields of influence, according to its
direction. These two varieties of current could be turned one into
the other by taking one of the rings up, turning it over, and putting
it down again in the plane. This operation is impossible to the plane
being, hence in this case also there would be two irreconcilable fields
in the plane. Now, if the plane being found two such irreconcilable
fields and could prove that they could not be accounted for by currents
in the rings, he would have to admit the existence of currents round
the rings—that is, in rings strung on the primary ring. Thus he would
come to admit the existence of a three-dimensional motion, for such a
disposition of currents is in three dimensions.
Now in our space there are two fields of different properties, which
can be produced by an electric current flowing in a closed circuit or
ring. These two fields can be changed one into the other by reversing
the currents, but they cannot be changed one into the other by any
turning about of the rings in our space; for the disposition of the
field with regard to the ring itself is different when we turn the
ring, over and when we reverse the direction of the current in the ring.
As hypotheses to explain the differences of these two fields and their
effects we can suppose the following kinds of space motions:—First, a
current along the conductor; second, a current round the conductor—that
is, of rings of currents strung on the conductor as an axis. Neither of
these suppositions accounts for facts of observation.
Hence we have to make the supposition of a four-dimensional motion.
We find that a four-dimensional rotation of the nature explained in a
subsequent chapter, has the following characteristics:—First, it would
give us two fields of influence, the one of which could be turned into
the other by taking the circuit up into the fourth dimension, turning
it over, and putting it down in our space again, precisely as the two
kinds of fields in the plane could be turned one into the other by a
reversal of the current in our space. Second, it involves a phenomenon
precisely identical with that most remarkable and mysterious feature of
an electric current, namely that it is a field of action, the rim of
which necessarily abuts on a continuous boundary formed by a conductor.
Hence, on the assumption of a four-dimensional movement in the region
of the minute particles of matter, we should expect to find a motion
analogous to electricity.
Now, a phenomenon of such universal occurrence as electricity cannot be
due to matter and motion in any very complex relation, but ought to be
seen as a simple and natural consequence of their properties. I infer
that the difficulty in its theory is due to the attempt to explain a
four-dimensional phenomenon by a three-dimensional geometry.
In view of this piece of evidence we cannot disregard that afforded
by the existence of symmetry. In this connection I will allude to the
simple way of producing the images of insects, sometimes practised by
children. They put a few blots of ink in a straight line on a piece of
paper, fold the paper along the blots, and on opening it the lifelike
presentment of an insect is obtained. If we were to find a multitude
of these figures, we should conclude that they had originated from a
process of folding over; the chances against this kind of reduplication
of parts is too great to admit of the assumption that they had been
formed in any other way.
The production of the symmetrical forms of organised beings, though not
of course due to a turning over of bodies of any appreciable size in
four-dimensional space, can well be imagined as due to a disposition in
that manner of the smallest living particles from which they are built
up. Thus, not only electricity, but life, and the processes by which we
think and feel, must be attributed to that region of magnitude in which
four-dimensional movements take place.
I do not mean, however, that life can be explained as a
four-dimensional movement. It seems to me that the whole bias of
thought, which tends to explain the phenomena of life and volition, as
due to matter and motion in some peculiar relation, is adopted rather
in the interests of the explicability of things than with any regard to
probability.
Of course, if we could show that life were a phenomenon of motion, we
should be able to explain a great deal that is at present obscure. But
there are two great difficulties in the way. It would be necessary to
show that in a germ capable of developing into a living being, there
were modifications of structure capable of determining in the developed
germ all the characteristics of its form, and not only this, but of
determining those of all the descendants of such a form in an infinite
series. Such a complexity of mechanical relations, undeniable though
it be, cannot surely be the best way of grouping the phenomena and
giving a practical account of them. And another difficulty is this,
that no amount of mechanical adaptation would give that element of
consciousness which we possess, and which is shared in to a modified
degree by the animal world.
In those complex structures which men build up and direct, such as a
ship or a railway train (and which, if seen by an observer of such a
size that the men guiding them were invisible, would seem to present
some of the phenomena of life) the appearance of animation is not due
to any diffusion of life in the material parts of the structure, but to
the presence of a living being.
The old hypothesis of a soul, a living organism within the visible one,
appears to me much more rational than the attempt to explain life as a
form of motion. And when we consider the region of extreme minuteness
characterised by four-dimensional motion the difficulty of conceiving
such an organism alongside the bodily one disappears. Lord Kelvin
supposes that matter is formed from the ether. We may very well suppose
that the living organisms directing the material ones are co-ordinate
with them, not composed of matter, but consisting of etherial bodies,
and as such capable of motion through the ether, and able to originate
material living bodies throughout the mineral.
Hypotheses such as these find no immediate ground for proof or disproof
in the physical world. Let us, therefore, turn to a different field,
and, assuming that the human soul is a four-dimensional being, capable
in itself of four dimensional movements, but in its experiences through
the senses limited to three dimensions, ask if the history of thought,
of these productivities which characterise man, correspond to our
assumption. Let us pass in review those steps by which man, presumably
a four-dimensional being, despite his bodily environment, has come to
recognise the fact of four-dimensional existence.
Deferring this enquiry to another chapter, I will here recapitulate the
argument in order to show that our purpose is entirely practical and
independent of any philosophical or metaphysical considerations.
If two shots are fired at a target, and the second bullet hits it
at a different place to the first, we suppose that there was some
difference in the conditions under which the second shot was fired
from those affecting the first shot. The force of the powder, the
direction of aim, the strength of the wind, or some condition must
have been different in the second case, if the course of the bullet
was not exactly the same as in the first case. Corresponding to every
difference in a result there must be some difference in the antecedent
material conditions. By tracing out this chain of relations we explain
nature.
But there is also another mode of explanation which we apply. If we ask
what was the cause that a certain ship was built, or that a certain
structure was erected, we might proceed to investigate the changes in
the brain cells of the men who designed the works. Every variation in
one ship or building from another ship or building is accompanied by
a variation in the processes that go on in the brain matter of the
designers. But practically this would be a very long task.
A more effective mode of explaining the production of the ship or
building would be to enquire into the motives, plans, and aims of the
men who constructed them. We obtain a cumulative and consistent body of
knowledge much more easily and effectively in the latter way.
Sometimes we apply the one, sometimes the other mode of explanation.
But it must be observed that the method of explanation founded on
aim, purpose, volition, always presupposes a mechanical system on
which the volition and aim works. The conception of man as willing and
acting from motives involves that of a number of uniform processes of
nature which he can modify, and of which he can make application. In
the mechanical conditions of the three-dimensional world, the only
volitional agency which we can demonstrate is the human agency. But
when we consider the four-dimensional world the conclusion remains
perfectly open.
The method of explanation founded on purpose and aim does not, surely,
suddenly begin with man and end with him. There is as much behind the
exhibition of will and motive which we see in man as there is behind
the phenomena of movement; they are co-ordinate, neither to be resolved
into the other. And the commencement of the investigation of that will
and motive which lies behind the will and motive manifested in the
three-dimensional mechanical field is in the conception of a soul—a
four-dimensional organism, which expresses its higher physical being
in the symmetry of the body, and gives the aims and motives of human
existence.
Our primary task is to form a systematic knowledge of the phenomena
of a four-dimensional world and find those points in which this
knowledge must be called in to complete our mechanical explanation of
the universe. But a subsidiary contribution towards the verification
of the hypothesis may be made by passing in review the history of
human thought, and enquiring if it presents such features as would be
naturally expected on this assumption.