There is another argument, one relating to the theories of time
and space, which greatly affects the conception of omnipresence.
This is the argument of the many dimensions, called by Keyser the
“radiant concept of hyper-space, which only a generation ago was
regarded, even by mathematicians--most adventurous of men--as being
purposeless and vain, but which meanwhile has advanced so rapidly to
commanding position that even the following statement by Poincaré,
in his recent address before the International Mathematical Congress
at Rome on ‘L’Avenir des Mathématiques,’ is well within the limits
of conservatism: ‘Nous sommes aujourd’hui tellement familiarisés
avec cette notion que nous pouvons en parler, même dans un cours
d’université, sans provoquer trop d’étonnement.’ The fact is that the
doctrine already exists in a vast and rapidly growing literature,
flourishes in all the scientific languages of the world, and in its
essential principles has become for mathematics as orthodox as the
multiplication table.”
The present position of the theory is briefly this: If there did
not exist a fourth dimension, we could not be aware of a third as
such, and so on. Are we then looking out upon a third dimensional
world, and realising it as such because we are mentally capable of
conceiving dimensions beyond it? Our world sensibly contains one
dimensional and two dimensional facts--the first such as a time
series, for which one number is sufficient to fix a point, and the
second such as a plane where position can be fixed by two numbers.
Does our world contain facts of other dimensions?
“All particles of air are four-dimensional in magnitude when,
in addition to their position in space, we also consider the
variable densities which they assume, as they are expressed
by the different heights of the barometer in the different
parts of the atmosphere. Similarly all conceivable spheres in
space are four-dimensional magnitudes, for their centres form
a three-dimensional point-aggregate, and around each centre a
one-dimensional totality of spheres, the radii of which can be
expressed by every numerical magnitude from zero to infinity.
Further, if we imagine a measuring-stick of invariable length to
assume every conceivable position in space, the positions so obtained
will constitute a five-dimensional aggregate. For in the first place
one of the extremities of the measuring-stick may be conceived to
assume a position at every point of space, and this determines for
one extremity alone of the stick a three-dimensional totality of
position, and, secondly, as we have seen above, there proceeds from
every such position of this extremity a two-dimensional totality
of directions, and by conceiving the measuring-stick to be placed
lengthwise in every one of these directions, we shall obtain all the
conceivable positions which the second extremity can assume, and
consequently the dimensions must be 3 + 2 or 5 …” &c., &c.[30]
Mathematicians have for long done problems in the seventh and eighth
dimensions. They have told us that you cannot tie a knot in the
second dimension, because there is no up or down, and the threads
would not cross--nor in the fourth, because the knot would pull out
in a new direction and would not hold. But it has only lately been
realised that fourth and other dimensions may be actual fact in the
world round us. Of course, from the point of view of a point there
are only three dimensions to be known, but to a line in the same
space there are five, to the surface probably six. Our intelligence
at present does not go beyond the point; but if we could think of
space from the point of view of a solid, worlds upon worlds would
rise before our view.
Of the fourth dimension we can discover some facts by analogy. We
can count the edges of its typical figure, and apply thought to
determining some of its conditions. But a more interesting subject
of research is the inquiry into the light thrown by the theory of
four dimensions on the determination of certain atoms in chemistry,
that are known to be distinct elements, but could only be determined
actually in another dimension.[31]
“In chemistry, the molecules of a compound body are said to consist
of the atoms of the elements which are contained in the body, and
these are supposed to be situated at certain distances from one
another and to be held in their relative positions by certain
forces. At first the centres of the atoms were conceived to lie in
one and the same plane. But Wislicenus was led by researches in
paralactic acid to explain the differences of isomeric molecules of
the same structural formulæ by the different positions of the atoms
in _space_. In fact, four points can always be so arranged in space
that every two of them may have any distance from each other; and the
change of one of the six distances does not necessarily involve the
alteration of any other.
“But suppose our molecule consists of five atoms? Four of these
may be so placed that the distance between any two of them can be
made what we please. But it is no longer possible to give the fifth
atom a position such that each of the four distances by which it
is separated from the other atoms may be what we please. On the
contrary, the fourth distance is dependent on the three remaining
distances, for the space of experience has only three dimensions. If,
therefore, I have a molecule which consists of five atoms, I cannot
alter the distance between two of them without at least altering some
second distance. But if we imagine the centres of the atoms placed
in a four-dimensioned space, this can be done; all the ten distances
which may be conceived to exist between the five points will then be
independent of one another. To reach the same result in the case of
six atoms we must assume a five-dimensional space, and so on.”[32]
Here we see that if chemistry as a science is bound to take account
of all its facts, the scientist is confronted with a problem of
dimensions that is really a problem of Infinity applied not, as in
the other cases quoted, to number, but to space.
And there is a reason which explains why the same problem tends
to appear in these different ways. Both time and space can be
most correctly thought of as _series_: the former known to us as
possessing one direction, though possibly involving more, and the
latter three, though possibly involving more. Time is not a thing nor
a condition, but it is the way in which we are enabled to apprehend
the relations of actions to one another. The assumption of the
Pragmatist, that a different date in history is a new condition which
might affect a chemical experiment, is meaningless, unless by that he
intends to say that at the different date new conditions prevailed.
The general conclusion of recent thought is then to establish
the Idealist position more strongly by an appeal to mathematical
argument. This argument is strengthened by finding at the present
time some support in scientific fact and experiment. The Idealist
therefore appeals to fact, and his position rests ultimately on a
truth which has its aspects of conformity with scientific experiment
and with logical or mathematical proof.