SPACE
168
SPACE
Space thus conceived avoids many of the difficulties
raised against its reality. But there still remain ques-
tions that have taxed the ingenuity of philosophers.
What is to be thought of the infinity of space, which
to many philosophers seems to be an indisputable
postulate? Here we must carefully distinguish the
two ideas to which we alluded above. Mathemati-
cians do not understand infinity in the same sense as
philosophers. The latter consider absolute infinity
as the plenitude of being, being itself; spatial infinity
for them can signify only plenitude of extension.
There are no hmits to an infinite space, nowhere can
there exist a definite relation to its extremities or even
to itself. It is impossible to add even mentally any-
thing to such extension, for it would be an absurdity
to conceive anything greater than infinite extension.
Mathematical infinity is something quite difi'erent.
It is not considered solely in relation to the being to
which it is attributed, but in relation to this being and
to the determinations of limits possible to the intel-
lect. Whatever by its nature surpasses all the hmits
we can assign it, that is mathematically infinite. It
must be carefully noted that these two ideas in no
way coincide, since it is possible that the intellect may
not grasp the nature of a being fully enough to deter-
mine its limits: the possibihty that this nature may
surpass all assignable hmits does not involve the con-
clusion that the being is in itself unhmited. Mathe-
matical infinity introduces into the problem a factor
extrinsic to the nature of the being: the relative per-
fection, or rather the imperfection, of the human idea;
and it is noteworthy that in all problems concerning
quantity our intellect is, to a very great extent, de-
pendent on our senses and our imagination. This
distinction being established, we may remark that real
space evidently surpasses all that experience can teach
us. We are forced, consequently, to solve the prob-
lem by analysis.
Mathematical space is abstract and mathematically infinite; but we are dealing here with the real universe. The notion of mathematical infinity may be apphed to it in a secondary sense. The nature of real space is such as not to demand any definite dimensions. No part of space in itself needs be the last. For all we know, or do not know, about it, space may be greater than any limits whatsoever we might assign. But space cannot be metaphysically infinite. It is impos- sible to have an actual quantitive infinite being com- posed of finite parts. To infinite extension nothing can be added, and from it nothing can be taken away, even mentally. For if, by hypothesis, infinite exten- sion is divided in two, neither of the parts is infinite since neither by itself contains the plentitude of ex- tension. Both therefore are finite; by their union they would form the original whole, but it is absurd to imagine that an infinite whole is formed by the union of two finite parts. It is clear that we can mentally take way a portion of space. Hence it is clear that space cannot be metaphysically infinite. An actu- ally infinite quantity is a contradiction in terms. Here of course our imagination cannot follow our in- tellect. We cannot represent exactly to ourselves what may be the limits of the world; and it is clear that in this case certain physical laws, those of mo- tion, for instance, cannot be fully apphed. It is use- less to discuss the subject further because, owing to the limitations of our experience, we are apt to indulge in mere fantastic and arbitrary speculations.
A stiU more abstruse subject is reached when we come to deal with the number of dimensions of space and its homogeneity. Our imagination always rep- resents real .space as having but three dimensions. We reach this intuitive space (see below) spoiila- neously; it s(^('ms to us so natural, so inevitable, that we have great difficulty in freeing ourselves from the domination of this image, and in conceiving (to imagine it is impossible) a space with more than three
dimensions. However, the question has been raised;
for geometricians reason frequently about a space of
four, of five, or of n dimensions. The problem is not
of the experimental order. Our sensory experiences
and everything in practical hfe reveal only three
dimensions. But does experience e.xhaust the possi-
bihties of real space? and can this space have no more
than three dimensions? Nothing obhges us to beheve
that such is the case. The material world requires
essentially only quantity, and this is not identical with
extension. Quantity confers on substance a multi-
plicity of parts; extension supposes this multiplicity
and gives a relative position to the parts. Quantity
implies a distinction of parts, extension adds extrapo-
sition, i. e. the placing of part outside of part; hence it
will be seen that, in a strict sense, material beings do
not necessarily postulate extension. It would then
be quite arbitrary to declare a priori that they must
have extension according to three mutually perpen-
dicular directions, and that they cannot have any
more. The word dimensions is here used, of course,
only by analogy with the thi-ee dimensions perceived
by experience; we can get at pure quantity only
through extension. But the intellect in its analysis
goes beyond the data offered to it by sense, and it is
forced to conclude that space of more than three
dimensions implies no contradiction.
By a very similar process we can solve the problem, so perplexing for the a\-erage mind, of the homo- geneity of space. The essential properties of quantity require no definite number of dimensions. The same may be said of the quahty, or rather intensity, of extension: the parts may be more or less extraposed. The parts, remaining the same, may give a greater or a less extension in the ordinary sense of the word. There is nothing contradictory, therefore, in all the parts of space being everj-where equally extraposed, in which case space would be homogeneous. But, on the other hand, there is no reason why space should not be differently extraposed in difi'erent parts, and if this be so, space would be heterogeneous; and if the variation be simple and constant, we can formulate the laws of these spaces and determine the properties of the figures formed therein. This explains why geometry, so rigorous in its methods and simple in its postulates, is not necessarily one. The ancient geom- etry of Euclid takes for granted the homogeneity of space; but it is weD known that non-Euclidian geome- tries have been constructed, notably those of Riemann and of Lobatchew.ski, difTering from Euclid's and yet free from all incoherency.
These speculations on the nature of space cannot, however, do away with the fundamental fact that the human mind is dominated by an image, imposing ir- resistibly on it a homogeneous tri-dimensional space. One of the central questions of classic psychology con- cerns the origin of this representation. We dismiss Kant's well-known view, that space is an a priori form of sensory activity. But psychologists fluctuate be- tween two extremes: on the one hand, nativism, rep- resented by Johann Muller, Fichte. Sigwart, Much, and many others; and on the other hand, empiricism, followed by Locke, Hume, Condillac, Maine de Biran, John Stuiirt Mill, Bain, Spencer, and others. The former hold that we obtain the image of space from the primordial subjective dispositions of our mental- ity; and many of them see therein a condition prece- dent of all experience. The second class, on the con- trary, believe that this image is acquiied, that it results from visual and tactih^ impressions and is only a result of association. Many authorit ies hesitate and try to discover an intermediate position. From the facts adduced and the analysis to which they have ln'i'U subjecti'd it seems clear that the image of space is in reality acquired like all other images: in very young children we see it, so to say, in process of form- ation. It is the result of the spontaneous interpreta-
