the application of the idea of multiple-dimensioned space to theoretical
chemistry. (See the treatise before cited.) 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_. (Compare “La chimie dans l’espace” by van’t Hoff,
1875, preface by J. Wislicenus). 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. Quite 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.
Now, if the independence of all the possible distances between the atoms
of a molecule is absolutely required by theoretical chemical research,
the science is really compelled, if it deals with molecules of more
than four atoms, to make use of the idea of a space of more than three
dimensions. This idea is, in this case, simply an instrument of research,
just as are, also, the ideas of molecules and atoms—means designed to
embrace in an obvious and systematic form the phenomena of chemistry
and to discover the conditions under which new phenomena can be evoked.
Whether a four-dimensioned space really exists is a question whose
insolubility cannot prevent research from making use of the idea, exactly
as chemistry has not been prevented from making use of the notion of
atom, although no one really knows whether the things we call atoms exist
or not.
IV.
REFUTATION OF THE ARGUMENTS ADDUCED TO PROVE THE EXISTENCE OF A
FOUR-DIMENSIONED SPACE INCLUSIVE OF THE VISIBLE WORLD.
The considerations of the preceding section will have convinced
the cultured non-mathematician of the service which the theory of
multiple-dimensioned spaces has done, and bids fair to do, for
geometrical research. In addition thereto is the consideration that every
extension of one branch of mathematical science is a constant source of
beneficial and helpful influence to the other branches. The knowledge,
however, that mathematicians can employ the notion of four-dimensioned
space with good results in their researches, would never have been
sufficient to procure it its present popularity; for every man of
intelligence has now heard of it, and, in jest or in earnest, often
speaks of it. The knowledge of a four-dimensioned space did not reach
the ears of cultured non-mathematicians until the consequences which the
spiritualists fancied it was permissible to draw from this mathematical
notion were publicly known. But it is a tremendous step from the
four-dimensioned space of the mathematicians to the space from which the
spirit-friends of the spiritualistic mediums entertain us with rappings,
knockings, and bad English. Before taking this step we will first discuss
the question of the real existence of a four-dimensional space, not
judging the question whether this space, if it really does exist, is
inhabited by reasonable beings who consciously act upon the world in
which we exist.
[Illustration: Fig. 4.]
Among the reasons which are put forward to prove the existence of a
four-dimensional space containing the world, the least reprehensible are
those which are based on the existence of symmetry. We spoke above of
two triangles in the same plane which have all their sides and angles
congruent, but which cannot be made to coincide by simple displacement
within the plane; but we saw that this coincidence could be effected by
holding fast one side of one triangle and moving it out of its plane
until it had been so far turned round that it fell back into its plane.
Now something similar to this exists in space. Cut two figures, exactly
like that of Fig. 4., out of a piece of paper, and turn the triangle
_ABF_ about the side _AB_, _ACE_ about the side _AC_, _BCD_ about the
side _BC_, and in one figure above and in the other below; then in both
cases the points _D_, _E_, _F_ will meet at a point, because _AE_ is
equal to _AF_, _BF_ is equal to _BD_, _CD_ is equal to _CE_. In this
manner we obtain two pyramids which in all lengths and all angles are
congruent, yet which cannot, no matter how we try, be made to coincide,
that is, be so fitted the one into the other that they shall both stand
as one pyramid. But the _reflected_ image of the one could be brought
into coincidence with the other. Two spatial structures whose sides and
angles are thus equal to each other, and of which each may be viewed
as the reflected image of the other, are called _symmetrical_. For
instance, the right and the left hand are symmetrical; or, a right and
a left glove. Now just as in two dimensions it is impossible by simple
displacement to bring into congruence triangles which like those above
mentioned can only be made to coincide by circumversion, so also in three
dimensions it is impossible to bring into congruence two symmetrical
pyramids. Careful mathematical reflection, however, declares, that this
could be effected if it were possible, while holding one of the surfaces,
to move the pyramid out of the space of experience, and to turn it round
through a four-dimensioned space until it reached a point at which it
would return again into our experiential space. This process would simply
be the four-dimensional analogon of the three-dimensional circumversion
in the above-mentioned case of the two triangles. Further, the interior
surfaces in this process would be converted into exterior surfaces,
and _vice versa_, exactly as in the circumversion of a triangle the
anterior and posterior sides are interchanged. If the structure which is
to be converted into its symmetrical counterpart is made of a flexible
material, the interchange mentioned of the interior and exterior surfaces
may be effected by simply turning the structure inside out; for example,
a right glove may thus be converted into a left glove.
Now from this truth, that every structure can be converted by means
of a four-dimensional space inclusive of the world, into a structure
symmetrical with it, it has been sought to establish the probability of
the real existence of a four-dimensioned space. Yet it will be evident,
from the discussions of the preceding section, that the only inference
which we can here make is, that the idea of a four-dimensioned space is
competent, from a mathematical point of view, to throw some light upon
the phenomena of symmetry. To conclude from these facts that a space
of this kind really exists, would be as daring as to conclude from
the fact that the uniform angular velocity of the apparent motions of
the fixed stars is explicable from the assumption of an axial motion
of the firmament, that the fixed stars are really rigidly placed in a
celestial sphere rotating about its axis. It must not be forgotten that
our comprehension of the phenomena of the real world consists of two
elements: first, of that which the things really are; and, second, of
that by which we rationally apprehend the things. This latter element
is partly dependent on the sum of the experiences which we have before
acquired, and partly on the necessity, due to the imperfection of reason,
of our embracing the multitudinous isolated phenomena of the world into
categories which we ourselves have formed, and which, therefore, are not
wholly derived from the phenomena themselves, but to a great extent are
dependent on us.
Besides geometrical reasons, Zöllner has also adduced cosmological
reasons to prove the existence of a four-dimensional space. To these
reasons belong especially the questions whether the number of the fixed
stars is infinitely great, whether the world is finite or infinite
in extension, whether the world had a beginning or will have an end,
whether the world is not hastening towards a condition of equilibrium or
dead level by the universal distribution of its matter and energy; the
problems, also, of gravitation and action at a distance; and finally, the
questions concerning the relations between the phenomena in the world of
sense-perception to the unknown things-in-themselves. All these questions
which can be decided in no definite sense, led Zöllner and his followers
to the assumption that a four-dimensioned space inclusive of the space of
experience must really exist. But more careful reflection will show that
this assumption does not dispose of the difficulties but simply displaces
them into another realm. Furthermore, even if four-dimensioned space did
unravel and make clear all the cosmological problems which have bothered
the human mind, still, its existence would not be proved thereby; it
would yet remain a mere hypothesis, designed to render more intelligible
to a being who can only make experiences in a three-dimensional space,
the phenomena therein which are full of mystery to it. A four-dimensioned
space would in such case possess for the metaphysician a value similar
to that which the ether possesses for the physicist. Still more
convincing than these cosmological reasons to the majority of men is
the physio-psychological reason drawn from the phenomena of vision which
Zöllner adduces. Into this main argument we will enter in more detail.
When we “see” an object, as we all know, the light which proceeds or is
reflected therefrom produces an image on the retina of our eye; this
image is conducted to our consciousness by means of the optical nerve,
and our reason draws therefrom an inference respecting the object. When,
now, we look at a square whose sides are a decimetre in length, and
whose centre is situated at the distance of a metre from the pupil of
our eye, an image is produced on the retina. But exactly the same image
will be produced there if we look at a square whose sides are parallel
to the sides of the first square but two decimetres in length, and whose
centre is situated at a distance of two metres from the pupil of the eye.
Proceeding thus further, we readily discover that an eye can perceive
in any length or line only the ratio of its magnitude to the distance
at which it is situated from it, and that generally a three-dimensional
world must appear to the eye two-dimensional, because all points which
lie behind each other in the direction outwards from the eye produce
on the retina only one image. This is due to the fact that the retinal
images are themselves two-dimensional; for which reason, Zöllner says,
the world must appear to a child as two-dimensional, if it be supposed to
live in a primitive condition of unconscious mental activity. To such a
child two objects which are moving the one behind the other, must appear
as suffering displacement on a surface, which we conceive behind the
objects, and on which the latter are projected. In all these apparent
displacements, coincidences and changes of form also are effected. All
these things must appear puzzling to a human being in the first stages
of its development, and the mind thus finds itself, as Zöllner further
argues, in the first years of childhood forced to adopt a hypothesis
concerning the constitution of space and to assume that the world is
three-dimensional, although the eye can really perceive it as only
two-dimensional. Zöllner then further says, that in the explanation of
the effects of the external world, man constantly finds this hypothesis
of his childish years confirmed, and that in this way it has become in
his mind so profound a conviction that it is no longer possible for him
to think it away. Consonant with this argumentation, also, is Zöllner’s
remark, that the same phenomenon has presented itself in astronomical
methods of knowledge. To explain the movements of the planets, which
appear to describe regular paths on the surface of a celestial sphere,
we were compelled in the solution of the riddles which these motions
presented, to assume in the structure of the heavens a dimension of
“depth,” and the complicated motions in the two-dimensioned firmament
were converted into very simple motions in three-dimensioned space.
Zöllner also contends that our conception of the entire visible world as
possessed of three dimensions is a product of our reason, which the mind
was driven to form by the contradictions which would be presented to it
on the assumption of only two dimensions by the perspective distortions,
coincidences, and changes of magnitude of objects. When a child moves its
hand before its eyes, turns it, brings it nearer, or pushes it farther
away, this child successively receives the most various impressions on
the surface of its retina of one and the same object of whose identity
and constancy its feelings offer it a perfect assurance. If the child
regarded the changeable projection of the hand on the surface of the
retina as the real object, and not the hand which lies beyond it, the
child would constantly be met with contradictions in its experience,
and to avoid this it makes the hypothesis that the space of experience
is three-dimensional. Zöllner’s contention is, therefore, that man
originally had only a two-dimensional intuition of space, but was forced
by experience to represent to himself the objects which on the retinal
surface appeared two-dimensional, as three-dimensional, and thus to
transform his two-dimensional space-intuition into a three-dimensional
one. Now, in exactly the same way, according to Zöllner’s notion, will
man, by the advancement and increasing exactness of his knowledge of
the phenomena of the outer world, also be compelled to conceive of the
material world as a “shadow cast by a more real four-dimensional world,”
so that these conceptions will be just as trivial for the people of the
twentieth century as since Copernicus’s time the explanation of the
motions of the heavenly bodies by means of a three-dimensional motion
has been.
Zöllner’s arguments from the phenomena of vision may be refuted as
follows: In the first place it is incorrect to say that we see the
things of the external world by means of two-dimensional retinal images.
The light which penetrates the eye causes an irritation of the optical
nerves, and any such effect which, though it be not powerful, is,
nevertheless, a mechanical one, can only take place on things which are
material. But material things are always three-dimensional. The effect
of light on the sensitive plates of photography can with just as little
justice be regarded as two-dimensional. Our senses can have perception
of nothing but three-dimensional things, and this perception is effected
by forces which in their turn act on three-dimensional things, namely,
our sensory nerves. It is wrong to call an image two-dimensional,
for it is only by abstraction that we can conceive of a thickness so
growing constantly smaller and smaller as to admit of our regarding a
three-dimensional picture as two-dimensional, by giving it in mind a
vanishingly small thickness. It is also wrong to say, as Zöllner says,
that when we see the shadow of a hand which is cast upon a wall we see
something two-dimensional. What we really perceive is that no light
falls upon our eye from the region included by the shadow, while from
the entire surrounding region light does fall on our eye. But this light
is reflected from the material particles which form the surface of the
wall, that is, from three-dimensional particles of matter. We must
always remember that our eye communicates to us only three-dimensional
knowledge, and that for the comprehension of anything which has two, one,
or no dimensions, _a purely intellectual act of abstraction must be added
to the act of perception_. When we imagine we have made a lead-pencil
mark on paper, we have, exactly viewed, simply heaped along side of each
other little particles of graphite in such a manner that there are by
far fewer graphite particles in the lateral and upward directions than
there are in the longitudinal direction, and thus our reason arrives by
abstraction at the notion of a straight line. When we look at an object,
say a cube of wood, we recognise the object as three-dimensional, and
it is only by abstraction that we can conceive of its two-dimensional
surfaces, of its twelve one-dimensional edges, and of its eight
no-dimensional corners. For we reach the perception of its surface, for
example, solely by reason of the fact that the material particles which
form the cube prevent the transmission of light, and reflect it, whereby
a part of the light reflected from every material particle strikes our
eye. Now, by thinking exclusively of those material particles which are
reflected, in contrariety to the empty space without and the hidden and
therefore non-reflected particles within, we form the notion of a surface.
It is evident from this, that all that we perceive is three-dimensional,
that we cannot come at anything two-dimensional without an intellectual
abstraction, and that, therefore, we cannot conceive of anything
two-dimensional exerting effects upon material things. But this fact is a
refutation of the retinal argument of Zöllner. If vision consisted wholly
and exclusively in the creation of a two-dimensioned image, the things
which take place in the world could never come into our consciousness.
The child, therefore, does not originally apprehend the world, as Zöllner
says, as two-dimensional; on the contrary, it apprehends it either not
at all, or it apprehends it as three-dimensional. Of course the child
must first “learn how” to see. It is found from the observation of
children during the first months of their lives, and of the congenitally
blind, who have suddenly acquired the power of vision by some successful
operation, that seeing does not consist alone in the irritations which
arise in the optic nerves, but also in the correct interpretation of
these irritations by reason. This correct interpretation, however, can
be accomplished only by the accumulation of a considerable stock of
experience. Especially must the recognition of the distance of the object
seen, be gradually learned. In this, two things are especially helpful;
first, the fact that we have two eyes and, consequently, that we must
feel two irritations of the optic nerves which are not wholly alike; and
secondly, the fact that we are enabled by our power of motion and our
sense of touch to convince ourselves of the distance and form of the
bodies seen. The question now arises, what sort of an intuition of space
would a creature have that had only one eye, that could neither move
itself nor its eye, and also possessed no peripheral nerves. According
to Zöllner’s view, this creature could, owing to its two-dimensional
retinal images, only have a two-dimensional intuition of space. The
author’s opinion, however, is, that such a creature could not see at all,
as it has no possibility of collecting experiences which are adapted in
any way to interpreting the effects of things on its retina. The light
which proceeded from the objects roundabout and fell on the retina could
produce no other effect on the being than that of a wholly intelligible
irritation, or, perhaps, even pain.
The reflections presented sufficiently show that neither the phenomena
of symmetry nor the retinal images of the objects of vision necessarily
force upon us the assumption of a four-dimensioned space. If the
material world should ever present problems which could not in the
progress of knowledge be solved in a natural way, the assumption that
a four-dimensional space containing the world exists would also be
incompetent to resolve the difficulties presented; it would simply
convert these difficulties into others, and not dispose of the problems
but simply displace them to another world. Yet the question might be
asked, Is the existence of a four-dimensional space really _impossible_?
To answer this question, we must first clearly know what we mean by
“exist.” If existence means that the intellectual _idea_ of a thing
can be formed and that this idea shall not lead to contradictions with
other well-established ideas and with experience, we have only to
say that four-dimensioned space does exist, as the arguments adduced
in sections II and III have rendered plain. If, namely, the space
of four dimensions did not exist as a clear idea in the minds of
mathematicians, mathematicians could certainly not have been led by
this idea to results which are recognised by the senses as true, and
which really take place in our own representable space. But if existence
means “material actuality” we must say that we neither now nor in the
future can know anything about it. For we know material actuality
only as three-dimensional, our senses can only make three-dimensional
experiences, and the inferences of our reason, although they can
well abstract from material things, can never ascend to the point of
explaining a four-dimensional materiality. Just as little, therefore,
as we can locally fix the idea of a two-dimensional material world, as
little can we substantiate the notion of a four-dimensioned material
existence.
V.
EXAMINATION OF THE HYPOTHESIS CONCERNING THE EXISTENCE OF
FOUR-DIMENSIONAL SPIRITS.
In connection with the belief that the visible world is contained in a
four-dimensioned space, Zöllner and his adherents further hold that this
higher space is inhabited by intelligent beings who can act consciously
and at will on the human beings who live in experiential space. To invest
this opinion with greater strength Zöllner appealed to the fact that the
greatest thinkers of antiquity and of modern times were either wholly
of this opinion or at least held views from which his contentions might
be immediately derived. Plato’s dialogue between Socrates and Glaukon
in the seventh book of the Republic, is evidence, says Zöllner, that
this greatest philosopher of antiquity possessed some presentiment of
this extension of the notion of space. Yet any one who has connectedly
studied and understood Plato’s system of philosophy must concede that
the so-called “ideas” of the Platonic system denote something wholly
different from what Zöllner sees in them or pretends to see. Zöllner
says that these Platonic ideas are spatial objects of more than three
dimensions and represent “real existence” in the same sense that the
material world, as contrasted with the images on the retina, represents
it. Zöllner similarly deals with the Kantian “thing-in-itself,” which is
also regarded as an object of higher dimensions.