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world is neither dark nor light, neither sounding nor silent,
is fairly hard to realize. And we are not much helped in
the realization by being told that the things are really there,
only they are altogether different from what they appear.
" Transfigured realism " is a broken reed. The distinction
of primary and secondary qualities will not work. But if
we can have an experience of a common-to-all in sense, even
when there is no extra-mentality in the object, we might
equally have it in connection with spatial phenomena in

The real problem here divides into two. First, can we
have an experience of an order, or of thought contents and
relations, which shall be valid for all ? Secondly, how can
we have such experience ? The first is simply a question of
fact; and the answer must be in the affirmative. To the


second question no answer can be given. "We do not know
how we reach the common-to-all; we only know that we
reach it. This is the deep mystery which is involved in the
community of finite minds ; and its solution must finally be
sought in the realm of the infinite.

But to the second question common-sense thinks it gives
an answer. This illusion is due to picturing the object in
space with other bodies about it which represent the know-
ing subjects. With this image well in mind, it is easy to
see how they all have the same object ; for they are all gath-
ered round the object and everybody sees it to be one and
the same. But this delusive clearness disappears when
we remember the process of perception. We never can get
nearer the object than our thought will carry us ; and the
object exists for us as anything independent of our thought
only through the rational necessity we find of positing the
object as an independent and universal content. This ne-
cessit}' is the bottom fact in the case ; and it can be referred
to nothing else. But this is quite as possible with the ex-
perience of phenomena as with any other. The identity of
the object is not secured by having a real thing in a real
space, but only by its being a factor of that rational world
which is the meaning and substance of the phenomenal
world, and which is the presupposition of every theory of
knowledge which understands itself and its problem.

We have now to decide between the views of space. In
any case, space must be a principle of intuition. One fact,
which makes the objectivity of space so unquestionable to
un reflective thought, is that we have apparently an imme-
diate perception of its existence, so that our perception of
space is as direct and immediate as our perception of things.
On the other hand, it is made an objection to the subjective
theory that it implies a deal of mental mechanism and men-


tal activity of which we are totally unconscious. Both po-
sitions are worthless as arguments. The apparently imme-
diate perception of space is, in any case, the result of non-
spatial activities. The existence of space would not account
for its perception. We must in some way be affected by it.
But space itself does not act upon the mind ; only things do
that. Hence our knowledge of space is a mental interpre-
tation of the action of things upon the mind. In this ac-
tion, spatial properties are displaced by varying intensities
of activity, and these variations are translated by the mind
into space-terms. These considerations show that our space
intuition must in any case arise within, and that the objec-
tive space is no factor of sense perception whatever. There
is no need of the real space to explain our experience.

But we have further seen that the realistic view is in-
consistent, and upon analysis even unintelligible. It hovers
between making space something and nothing, and both
views are absurd. It also conflicts with the unity of being,
and forces us to regard the infinite as composed of parts.
Finally, it implies a hopeless dualism of first principles, in
that it implies the coexistence of two necessary and mutual-
ly independent principles. But this view is strictly impos-
sible, and any doctrine which leads to it must be rejected.
The attempt to regard space as a system of relations be-
tween things we found to be an impossible compromise be-
tween the subjective and the objective view. The objective
existence of space, then, is not only not proven, but it is in
itself unclear, inconsistent, and impossible. We reject it,
therefore, for the view that space is ultimately a principle
of intuition, and, secondarily, a mode of appearance. But
though subjective, it is not arbitrary or individual. A given
state of being may allow of only one space-translation, and
this translation may be universal and changeless in all intui-
tion, whether divine or human. However that may be, the


universe can have its spatial properties and relations only in
the mind, which not only belongs to the system, but is both
its foundation and its crown.

So, then, space is phenomenal. It is not a boundless void
in which things exist, but only the general form of objective
experience. But all that was ever true of it is true still ;
and the laws of space are as binding upon us as ever. We
cannot slip into the non-spatial and get about without mov-
ing. We may still go on making appointments to meet at
any given place, and there will be no obscurity about our
meaning. Within the phenomenal, space relations have the
clearest possible meaning. But when we abstract them from
things and set them up as realities by themselves, we are
" lost and embrangled in inextricable difficulties."

The relation of the infinite to space calls for brief men-
tion. We have affirmed that space, as the form of intuition,
may exist for the infinite as well as for the finite ; and this
may easily be mistaken for a limitation of the infinite. But
this would be to confound space as principle with space as
limitation. For human beings space has a double aspect.
It represents not only a principle of intuition, but also a
limitation of our agency. The organism which conditions
our mental activity has space relations, and thus we natu-
rally appear to be located and limited in space. But this
location is of the organism only, and this limitation is only
the result of our dynamic limitations. ' It consists solely in
the fact that our immediate action upon reality is limited.
Far and near are terms which depend entirely upon the
amount of mediation necessary to affect any given reality.
Wherever we act immediately, there we are ; so that, instead
of saying we can act only where we are, we ought rather to
say we are wherever we act. But our immediate action ex-
tends to only a few things, and this fact appears as spatial
limitation. In this sense of limitation, space cannot be af-


firmed of the infinite. It comprises all reality in the unity
of its immediate activity, and hence is everywhere. For by
omnipresence we can mean nothing more than this immedi-
ate action upon all reality. The conception of omnipresence
as a boundless space-filling bulk is a contradiction, for that
which is in space and fills space cannot be omnipresent in
space, but different parts must be in different places. Each
part, then, would be in its own place and nowhere else.
Thus the unity and omnipresence of the infinite would dis-

This modification of the spatial judgment by our organic
experience introduces a large element of relativity into it.
It is only the pure spatial judgment, as in geometry, which
can be regarded as universal. All beyond that is affected
by the general limitation of the finite and by our organic

Our general view of space can hardly fail to suggest the
much-debated question concerning the dimensions of space.
Of late years the claim has often been made by mathema-
ticians that space may not be restricted to three dimensions,
and elaborate discussions have been made of the properties
of non-Euclidian space. The most curious conclusions have
been drawn as to what would be true in such spaces, and
the impression has become very general that the conception
of space as having only three dimensions is mistaken. We
have now to inquire whether the principle of space is such
as necessarily to restrict it to three dimensions.

The principle of space has no such universality as the
laws of formal thought. These condition all our thinking,
but the principle of space conditions only our intuition of
objects. We must further allow that all forms of external
experience are not alike calculated to awaken the mind to
react with a spatialization of its objects. We must also ad-


mit that our nature may contain mysterious possibilities
which are at present entirely hidden. It is, then, possible
that, under certain forms of experience, the mind would
never come to the space intuition. It is equally possible
that, under other forms of sense-experience, the mind should
arrange its objects according to some altogether different
principle, so as to have a new form of intuition. This new
form, however, would not be space, but something quite
peculiar. As such, it would be related to the space-intui-
tion, as our sense of color is to that of sound. This, of
course, is a mere logical possibility, but there is certainly
no ground for saying that the space-intuition is the only one
possible in the nature of being. If there were any ground
for affirming the existence of such a new form, there would
be nothing a/priori incredible in it. It is entirely possible,
however, to hold, along with this admission, that the space-
intuition cannot be changed in its essential laws and nature.

In affirming that the dimensions of space are necessarily
three, and only three, it is important to premise that the
planes of reference are perpendicular each to the other two.
Without this assumption, the dimensions of space may be
as many as we please. But, with this assumption, the claim
is that the position of any point in space can be defined by
straight lines drawn to each of these planes of reference.
These straight lines are called the co-ordinates of the point,
and they tell us how far the point is from each of the planes.
The three planes represent the dimensions of space. Thus
far nothing has appeared in the affirmative which is not
purely hypothetical, or which does not confound the dimen-
sions of things in space with the dimensions of space itself.

The first class of arguments consists entirely of illustra-
tions drawn from analytic formulas. It is well known that
the formulas of analytics are independent of geometrical rep-
resentation. So far as the analytic reasoning goes, we are


free to choose n planes of reference, if we make no attempt
at spatial representation. These formulas, however, admit
of such representation when there are only three perpendic-
ular planes of reference ; and if n such planes were possible,
then a formula involving n planes would also be represent-
able. But this is far enough from proving that n planes
are possible; it only deduces a consequence from an as-

But there is no need to have recourse to elaborate
formulas to deduce this small conclusion. There is to the
uninitiated a certain air of mystery in an involved and
transcendental formula, and especially in a formula for a
" pseudo-spherical " surface, which may serve to impose on
the illogical mind, but the argument from such a formula
is in nothing better than the following : In algebra, a can
be represented by a line in space, #" by a plane surface, and
a" by a cube ; a* and all higher powers are unrepresentable.
So far as algebra is concerned, it is a mere coincidence that
<z, a', and a 3 are spatially representable, and the algebraic
analysis goes on in complete independence of space. It
deals with numbers and their relations, and these are log-
ical, and not spatial. But it would be quite easy to say
that, if space had n dimensions, then # n could be spatially
represented as well as a or # a or # 3 , and the argument would
be just as forcible as the mass of what is uttered on this
subject. In fact, mathematicians have fallen a prey to their
own terminology in this matter. Through desiring to give
the utmost generality to their analytic formulas, they have
constructed them without any regard to actual space. Then
they have discovered that, to make them representable, cer-
tain limitations must be made. Thus actual space is made
to appear as a special case; and this is called flat space,
Euclidian space, etc. But, by applying an adjective to
space, they have suggested to themselves the possibility of


other spaces, and forthwith any given set of analytic as-
sumptions passes for a space of the nth order. By this time
the illusion is complete, and the request for a proof that
those spaces of the nth order represent anything but ana-
lytic assumptions is resented as unkind.

The other class of arguments confounds the dimensions
of things in space with the dimensions of space itself. If
we omit reference to. the three perpendicular planes of ref-
erence, a thing may have any number of dimensions. The
various utterances concerning a curvature of space are all
instances of this confusion. What is meant by a curvature
of space itself is something which defies all comprehen-
sion, as much so as a curvature of number. It is assumed
that, in case of such curvature, straight lines would at last
return into themselves ; but the simple fact would be, not
that space is curved, but that the line is not straight, but
curved. This would be quite intelligible, while the doctrine
of a curved space is quite unintelligible. If it be said that
straight lines never occur in reality, we have no objection,
provided the claim be proved; but this is different from
affirming that truly straight lines are not straight, but
curved. The geometer does not assume anything about
the reality of lines, but contents himself with showing what
would be true of such lines, if they did exist. To determine
the content and implications of our space-intuitions is his
only aim ; and, knowing that these intuitions are purely
mental products, he is entirely free from doubts whether,
in some outlying regions of space, these principles may not
be invalid. Space being in the mind, and space-figures be-
ing mental constructions, they will always have the mean-
ing which the mind assigns to them, and hence can never
be twisted out of their proper significance.

This principle of a curvature of space has been invoked to
save the universe from finally running down. If space be


curved, then the outgoing energy will at last be restored,
and the system may keep agoing. But there is no need of
the unintelligible assumption of a curvature of space to ex-
press this result. We can simply say that, if the nature of
reality be such that radiant energy moves in curved lines,
then it will at last come back to the point of departure. Of
course, to make this assumption of any use, we should have
to make many others, but, such as it is, it is an attack, not on
our space-intuition, but on the first law of motion. In short,
all the illustrations of a space of n dimensions can be brought
into entire harmony with our space-intuition by substitut-
ing for a curvature of space a curvature in space, and for n
dimensions of space n dimensions of things in space. This
part of the doctrine seems to be largely due to the pestilent
practice of viewing straight lines as segments of circles
with an infinite radius. This custom, together with the
allied one of viewing parallel lines as meeting at an infinite
distance, has its practical advantage, but when it results in
confounding all definitions and in uttering complete non-
sense, it is high time to inquire whether the advantage be
not too dearly purchased.

A poor argument, however, though a suspicious circum-
stance, is not a disproof of the thing to be proved. The
doctrine of n dimensions can be tested only by a direct at-
tempt to realize its assumptions. Where, then, is the rath
dimension to be found ? One writer, in his explanation of
the disappearance of material bodies in spiritistic perform-
ances, assumes a fourth dimension of space, into which the
bodies are drawn by the spirits. If there were beings who
could observe only two dimensions of space, then a bod}^
which moved in the third dimension would disappear from
their vision. If, now, there be a fourth dimension, then the
spirits have only to draw the body into the fourth dimen-
sion to render it invisible. It would seem, then, that the


fourth dimension interpenetrates the three dimensions. The
solid body which disappeared was not out of the room, but
out of its three dimensions. And yet there was no point in
the room which could not be defined in a space of three
dimensions. The fourth dimension, therefore, is not some-
thing added to the three dimensions, but is something co-
incident with them ; that is, it is not a space-dimension at all,
but, if anything, it would be a state of matter in which it
would not appear in any way. The necessity of putting the
fourth dimension within the three dimensions deprives it of
all right to be called a dimension of space. Upon the whole,
it is not likely that the performances of sleight-of-hand
tricksters will contribute much to philosophic discovery.

The relation of the doctrine to geometry is not clearly
settled in the minds of its holders. Some would view it
simply as an extension of our present geometry, while
others would view it as an attack upon it. If we conceive
of beings dwelling in a plane and limited to conceptions of
lines in a plane, it is possible that such beings should form
a valid plane geometry ; and if afterwards the} 7 should ad-
vance to a conception of the third dimension of space, their
early geometry would be extended merely, and would be
as valid as ever. Now, in the same way, it may be claim-
ed that a new dimension of space would only extend our
present geometry without in any way discrediting it. In
that case the doctrine could be tested only by inquiring
whether the notion of a new dimension represents any-
thing more than a gratuitous assumption which defies all
construction and comprehension. But many holders of the
view regard it as conflicting with received geometry, and
this position makes it possible to test the view by reflecting
upon the character of geometrical truth. If that truth be
strictly true, then any doctrine which conflicts with it is
false. The believer in n dimensions will have to disprove


geometry before he can maintain his theory. If he insist
that straight lines return into themselves, that only shows
that he means by straight lines what others mean by curves.
If he claim that parallel lines may meet, it only shows that
he means by parallel lines what others mean by converging
lines. ISTor must he be allowed to make irrelevant appeals
to the nature of things, for geometry does not concern itself
with the nature of things, but with the nature and implica-
tions of our space-intuition.

A final word must be said concerning the unity of our
space-intuition. It is often assumed that there may be be-
ings which see things in only one or two dimensions, and
they would, of course, be as positive about the impossibility
of a third dimension as we are about a fourth. We know,
however, that they would be mistaken, and what better right
have we to insist on our view. If the fourth dimension be
assumed to contradict what we know of the three dimen-
sions, we should have the best right for rejecting it ; and
even if it were assumed only to extend our view, we should
have a right based on the unity of our space-intuition. For
these beings who see things only in one or two dimensions
are pure myths, and their possibility is far from apparent.
To begin with, the assumption that reality admits of any
number of space-intuitions falls back into the popular form
of Kantianism, according to which reality itself is quite in-
different to the forms of thought. But this is to divorce
thought and reality entirely, and to leave the thought with-
out any ground or explanation. But if reality is to explain
thought, then a given phase of reality admits only of a given
representation in thought. This notion that thought can
shift about and view reality in any and every way betrays
a total lack of appreciation of causation ; it is the supersti-
tion of a time which had no conception of law whatever.

Further, our intuition of space is not built up by adding


one dimension after another; but the first and second dimen-
sions are reached by abstracting from the unitary intuition
of a space of three dimensions. Given this intuition, it is
easy to attend to one dimension to the exclusion of the other
two ; but they could not be directly reached for the follow-
ing reasons : Suppose a being with an intuition of only one
dimension of space. At first we are tempted to think of
that one dimension as a line ; but this it could not be, because
to see it as a line, the being must be outside of the line, and
the line must be across the direction of vision. But this
would imply two dimensions of space the direction of the
line of vision and that of the line perceived. If we confine
him strictly to one dimension, the line must take the direc-
tion of the line of vision, and this would become a point.
But this point again could never be known as such, except
in relation to other points outside of the line, and as this is
contrary to the hypothesis, it could never be known as a
point at all. The line itself is without breadth or thickness,
and the being, if it knew itself as related to the line, must
know itself as in the line ; and all its other objects must be
in the line, and hence all alike must be known as without
breadth or thickness. For us who have the full space-
intuition, it is easy to abstract from two dimensions and
consider only the line, but for the being who has only the
one dimension the space-intuition would be impossible.

The same is true for the two dimensions. In this case
the being would be in a plane, but without any thickness.
He cannot rise above the plane to look at it, for this would
be to invoke the third dimension. He must stay then in
the surface, and must find all his objects in that surface.
But there can be no doubt that we are led to the conception
of a surface only by our experience with solids ; we reach it
by abstraction of the third dimension. If there were no
third dimension, we should certainly never have to come to


the notion of either line or surface. This being, however,
who is in the surface, and who knows nothing of any points
outside of the surface, would never know the surface at all.
The surface is conceivable only as a limit between different
parts of space, and, as these are impossible, the limit between
them is also impossible. We view our space -intuition as
properly a unit, and not as compounded of separate factors,
and these factors which we separate in thought are abstrac-
tions, which are possible only through the unity of space as
a form of three dimensions. All our dealing with the first
and second dimensions of space implies the three dimensions.
For the present, those who affirm that space may have n
dimensions must be judged either to be calling a series of
analytic assumptions by the misleading name of space or
else simply to be making a noise.



ACCORDING to the popular view, the world is in space and
has its history in time. "We have found ourselves compelled
to deny that the world is in space, for spatiality is only phe-
nomenal. We have next to inquire whether the world's
history in time is an ontological or only a phenomenal fact.
Kant made the same argument do for both space and time ;
but there are many difficulties in the case of time which do
not exist in that of space, and which compel a separate dis-
cussion. The subjectivity of time is by no means involved
in that of space. At the same time much that was said in
the previous chapter will apply here.

As in the case of space, we distinguish between the onto-
logical and the psychological question. We do not ask how
we come to the notion of time, but what it stands for after
we get it. Is it an existence, or a mode of existence, or only
a mode of our thinking ?

Kant set the example of calling space and time forms
of intuition, and this has led to a very general assumption

Online LibraryBorden Parker BowneMetaphysics → online text (page 13 of 34)