THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE
Parmenides, and the Asiatic thinkers with whom he is in close
affinity, propound a theory of existence which is in close accord
with a conception of a possible relation between a higher and a lower
dimensional space. This theory, prior and in marked contrast to the
main stream of thought, which we shall afterwards describe, forms a
closed circle by itself. It is one which in all ages has had a strong
attraction for pure intellect, and is the natural mode of thought for
those who refrain from projecting their own volition into nature under
the guise of causality.
According to Parmenides of the school of Elea the all is one, unmoving
and unchanging. The permanent amid the transient—that foothold for
thought, that solid ground for feeling on the discovery of which
depends all our life—is no phantom; it is the image amidst deception of
true being, the eternal, the unmoved, the one. Thus says Parmenides.
But how explain the shifting scene, these mutations of things!
“Illusion,” answers Parmenides. Distinguishing between truth and
error, he tells of the true doctrine of the one—the false opinion of a
changing world. He is no less memorable for the manner of his advocacy
than for the cause he advocates. It is as if from his firm foothold
of being he could play with the thoughts under the burden of which
others laboured, for from him springs that fluency of supposition and
hypothesis which forms the texture of Plato’s dialectic.
Can the mind conceive a more delightful intellectual picture than that
of Parmenides, pointing to the one, the true, the unchanging, and yet
on the other hand ready to discuss all manner of false opinion, forming
a cosmogony too, false “but mine own” after the fashion of the time?
In support of the true opinion he proceeded by the negative way of
showing the self-contradictions in the ideas of change and motion.
It is doubtful if his criticism, save in minor points, has ever been
successfully refuted. To express his doctrine in the ponderous modern
way we must make the statement that motion is phenomenal, not real.
Let us represent his doctrine.
[Illustration: Fig. 13.]
Imagine a sheet of still water into which a slanting stick is being
lowered with a motion vertically downwards. Let 1, 2, 3 (Fig. 13),
be three consecutive positions of the stick. A, B, C, will be three
consecutive positions of the meeting of the stick, with the surface of
the water. As the stick passes down, the meeting will move from A on to
B and C.
Suppose now all the water to be removed except a film. At the meeting
of the film and the stick there will be an interruption of the film.
If we suppose the film to have a property, like that of a soap bubble,
of closing up round any penetrating object, then as the stick goes
vertically downwards the interruption in the film will move on.
[Illustration: Fig. 14.]
If we pass a spiral through the film the intersection will give a point
moving in a circle shown by the dotted lines in the figure. Suppose
now the spiral to be still and the film to move vertically upwards,
the whole spiral will be represented in the film of the consecutive
positions of the point of intersection. In the film the permanent
existence of the spiral is experienced as a time series—the record
of traversing the spiral is a point moving in a circle. If now we
suppose a consciousness connected with the film in such a way that the
intersection of the spiral with the film gives rise to a conscious
experience, we see that we shall have in the film a point moving in a
circle, conscious of its motion, knowing nothing of that real spiral
the record of the successive intersections of which by the film is the
motion of the point.
It is easy to imagine complicated structures of the nature of the
spiral, structures consisting of filaments, and to suppose also that
these structures are distinguishable from each other at every section.
If we consider the intersections of these filaments with the film as it
passes to be the atoms constituting a filmar universe, we shall have in
the film a world of apparent motion; we shall have bodies corresponding
to the filamentary structure, and the positions of these structures
with regard to one another will give rise to bodies in the film moving
amongst one another. This mutual motion is apparent merely. The reality
is of permanent structures stationary, and all the relative motions
accounted for by one steady movement of the film as a whole.
Thus we can imagine a plane world, in which all the variety of motion
is the phenomenon of structures consisting of filamentary atoms
traversed by a plane of consciousness. Passing to four dimensions and
our space, we can conceive that all things and movements in our world
are the reading off of a permanent reality by a space of consciousness.
Each atom at every moment is not what it was, but a new part of that
endless line which is itself. And all this system successively revealed
in the time which is but the succession of consciousness, separate
as it is in parts, in its entirety is one vast unity. Representing
Parmenides’ doctrine thus, we gain a firmer hold on it than if we
merely let his words rest, grand and massive, in our minds. And we have
gained the means also of representing phases of that Eastern thought
to which Parmenides was no stranger. Modifying his uncompromising
doctrine, let us suppose, to go back to the plane of consciousness
and the structure of filamentary atoms, that these structures are
themselves moving—are acting, living. Then, in the transverse motion
of the film, there would be two phenomena of motion, one due to the
reading off in the film of the permanent existences as they are in
themselves, and another phenomenon of motion due to the modification of
the record of the things themselves, by their proper motion during the
process of traversing them.
Thus a conscious being in the plane would have, as it were, a
two-fold experience. In the complete traversing of the structure, the
intersection of which with the film gives his conscious all, the main
and principal movements and actions which he went through would be the
record of his higher self as it existed unmoved and unacting. Slight
modifications and deviations from these movements and actions would
represent the activity and self-determination of the complete being, of
his higher self.
It is admissible to suppose that the consciousness in the plane has
a share in that volition by which the complete existence determines
itself. Thus the motive and will, the initiative and life, of the
higher being, would be represented in the case of the being in the
film by an initiative and a will capable, not of determining any great
things or important movements in his existence, but only of small and
relatively insignificant activities. In all the main features of his
life his experience would be representative of one state of the higher
being whose existence determines his as the film passes on. But in his
minute and apparently unimportant actions he would share in that will
and determination by which the whole of the being he really is acts and
lives.
An alteration of the higher being would correspond to a different life
history for him. Let us now make the supposition that film after film
traverses these higher structures, that the life of the real being is
read off again and again in successive waves of consciousness. There
would be a succession of lives in the different advancing planes of
consciousness, each differing from the preceding, and differing in
virtue of that will and activity which in the preceding had not been
devoted to the greater and apparently most significant things in life,
but the minute and apparently unimportant. In all great things the
being of the film shares in the existence of his higher self as it is
at any one time. In the small things he shares in that volition by
which the higher being alters and changes, acts and lives.
Thus we gain the conception of a life changing and developing as a
whole, a life in which our separation and cessation and fugitiveness
are merely apparent, but which in its events and course alters,
changes, develops; and the power of altering and changing this whole
lies in the will and power the limited being has of directing, guiding,
altering himself in the minute things of his existence.
Transferring our conceptions to those of an existence in a higher
dimensionality traversed by a space of consciousness, we have an
illustration of a thought which has found frequent and varied
expression. When, however, we ask ourselves what degree of truth
there lies in it, we must admit that, as far as we can see, it is
merely symbolical. The true path in the investigation of a higher
dimensionality lies in another direction.
The significance of the Parmenidean doctrine lies in this that here, as
again and again, we find that those conceptions which man introduces of
himself, which he does not derive from the mere record of his outward
experience, have a striking and significant correspondence to the
conception of a physical existence in a world of a higher space. How
close we come to Parmenides’ thought by this manner of representation
it is impossible to say. What I want to point out is the adequateness
of the illustration, not only to give a static model of his doctrine,
but one capable as it were, of a plastic modification into a
correspondence into kindred forms of thought. Either one of two things
must be true—that four-dimensional conceptions give a wonderful power
of representing the thought of the East, or that the thinkers of the
East must have been looking at and regarding four-dimensional existence.
Coming now to the main stream of thought we must dwell in some detail
on Pythagoras, not because of his direct relation to the subject, but
because of his relation to investigators who came later.
Pythagoras invented the two-way counting. Let us represent the
single-way counting by the posits _aa_, _ab_, _ac_, _ad_, using these
pairs of letters instead of the numbers 1, 2, 3, 4. I put an _a_ in
each case first for a reason which will immediately appear.
We have a sequence and order. There is no conception of distance
necessarily involved. The difference between the posits is one of
order not of distance—only when identified with a number of equal
material things in juxtaposition does the notion of distance arise.
Now, besides the simple series I can have, starting from _aa_, _ba_,
_ca_, _da_, from _ab_, _bb_, _cb_, _db_, and so on, and forming a
scheme:
_da_ _db_ _dc_ _dd_
_ca_ _cb_ _cc_ _cd_
_ba_ _bb_ _bc_ _bd_
_aa_ _ab_ _ac_ _ad_
This complex or manifold gives a two-way order. I can represent it by
a set of points, if I am on my guard against assuming any relation of
distance.
[Illustration: Fig. 15.]
Pythagoras studied this two-fold way of counting in reference to
material bodies, and discovered that most remarkable property of the
combination of number and matter that bears his name.
The Pythagorean property of an extended material system can be
exhibited in a manner which will be of use to us afterwards, and which
therefore I will employ now instead of using the kind of figure which
he himself employed.
Consider a two-fold field of points arranged in regular rows. Such a
field will be presupposed in the following argument.
[Illustration: Fig. 16. 1 and 2]
It is evident that in fig. 16 four of the points determine a square,
which square we may take as the unit of measurement for areas. But we
can also measure areas in another way.
Fig. 16 (1) shows four points determining a square.
But four squares also meet in a point, fig. 16 (2).
Hence a point at the corner of a square belongs equally to four
squares.
Thus we may say that the point value of the square shown is one point,
for if we take the square in fig. 16 (1) it has four points, but each
of these belong equally to four other squares. Hence one fourth of each
of them belongs to the square (1) in fig. 16. Thus the point value of
the square is one point.
The result of counting the points is the same as that arrived at by
reckoning the square units enclosed.
Hence, if we wish to measure the area of any square we can take the
number of points it encloses, count these as one each, and take
one-fourth of the number of points at its corners.
[Illustration: Fig. 17.]
Now draw a diagonal square as shown in fig. 17. It contains one point
and the four corners count for one point more; hence its point value is
2. The value is the measure of its area—the size of this square is two
of the unit squares.
Looking now at the sides of this figure we see that there is a unit
square on each of them—the two squares contain no points, but have four
corner points each, which gives the point value of each as one point.
Hence we see that the square on the diagonal is equal to the squares
on the two sides; or as it is generally expressed, the square on the
hypothenuse is equal to the sum of the squares on the sides.
[Illustration: Fig. 18.]
Noticing this fact we can proceed to ask if it is always true. Drawing
the square shown in fig. 18, we can count the number of its points.
There are five altogether. There are four points inside the square on
the diagonal, and hence, with the four points at its corners the point
value is 5—that is, the area is 5. Now the squares on the sides are
respectively of the area 4 and 1. Hence in this case also the square
on the diagonal is equal to the sum of the square on the sides. This
property of matter is one of the first great discoveries of applied
mathematics. We shall prove afterwards that it is not a property of
space. For the present it is enough to remark that the positions in
which the points are arranged is entirely experimental. It is by means
of equal pieces of some material, or the same piece of material moved
from one place to another, that the points are arranged.
Pythagoras next enquired what the relation must be so that a square
drawn slanting-wise should be equal to one straight-wise. He found that
a square whose side is five can be placed either rectangularly along
the lines of points, or in a slanting position. And this square is
equivalent to two squares of sides 4 and 3.
Here he came upon a numerical relation embodied in a property of
matter. Numbers immanent in the objects produced the equality so
satisfactory for intellectual apprehension. And he found that numbers
when immanent in sound—when the strings of a musical instrument were
given certain definite proportions of length—were no less captivating
to the ear than the equality of squares was to the reason. What wonder
then that he ascribed an active power to number!
We must remember that, sharing like ourselves the search for the
permanent in changing phenomena, the Greeks had not that conception of
the permanent in matter that we have. To them material things were not
permanent. In fire solid things would vanish; absolutely disappear.
Rock and earth had a more stable existence, but they too grew and
decayed. The permanence of matter, the conservation of energy, were
unknown to them. And that distinction which we draw so readily between
the fleeting and permanent causes of sensation, between a sound and
a material object, for instance, had not the same meaning to them
which it has for us. Let us but imagine for a moment that material
things are fleeting, disappearing, and we shall enter with a far better
appreciation into that search for the permanent which, with the Greeks,
as with us, is the primary intellectual demand.
What is that which amid a thousand forms is ever the same, which we can
recognise under all its vicissitudes, of which the diverse phenomena
are the appearances?
To think that this is number is not so very wide of the mark. With
an intellectual apprehension which far outran the evidences for its
application, the atomists asserted that there were everlasting material
particles, which, by their union, produced all the varying forms and
states of bodies. But in view of the observed facts of nature as
then known, Aristotle, with perfect reason, refused to accept this
hypothesis.
He expressly states that there is a change of quality, and that the
change due to motion is only one of the possible modes of change.
With no permanent material world about us, with the fleeting, the
unpermanent, all around we should, I think, be ready to follow
Pythagoras in his identification of number with that principle which
subsists amidst all changes, which in multitudinous forms we apprehend
immanent in the changing and disappearing substance of things.
And from the numerical idealism of Pythagoras there is but a step to
the more rich and full idealism of Plato. That which is apprehended by
the sense of touch we put as primary and real, and the other senses we
say are merely concerned with appearances. But Plato took them all as
valid, as giving qualities of existence. That the qualities were not
permanent in the world as given to the senses forced him to attribute
to them a different kind of permanence. He formed the conception of a
world of ideas, in which all that really is, all that affects us and
gives the rich and wonderful wealth of our experience, is not fleeting
and transitory, but eternal. And of this real and eternal we see in the
things about us the fleeting and transient images.
And this world of ideas was no exclusive one, wherein was no place
for the innermost convictions of the soul and its most authoritative
assertions. Therein existed justice, beauty—the one, the good, all
that the soul demanded to be. The world of ideas, Plato’s wonderful
creation preserved for man, for his deliberate investigation and their
sure development, all that the rude incomprehensible changes of a harsh
experience scatters and destroys.
Plato believed in the reality of ideas. He meets us fairly and
squarely. Divide a line into two parts, he says; one to represent
the real objects in the world, the other to represent the transitory
appearances, such as the image in still water, the glitter of the sun
on a bright surface, the shadows on the clouds.
A B
——————————————————————————————|————————————————————————————————-
Real things: Appearances:
_e.g._, the sun. _e.g._, the reflection of the sun.
Take another line and divide it into two parts, one representing
our ideas, the ordinary occupants of our minds, such as whiteness,
equality, and the other representing our true knowledge, which is of
eternal principles, such as beauty, goodness.
A^1 B^1
——————————————————————————————|————————————————————————————————-
Eternal principles, Appearances in the mind,
as beauty as whiteness, equality
Then as A is to B, so is A^1 to B^1
That is, the soul can proceed, going away from real things to a region
of perfect certainty, where it beholds what is, not the scattered
reflections; beholds the sun, not the glitter on the sands; true being,
not chance opinion.
Now, this is to us, as it was to Aristotle, absolutely inconceivable
from a scientific point of view. We can understand that a being is
known in the fulness of his relations; it is in his relations to his
circumstances that a man’s character is known; it is in his acts under
his conditions that his character exists. We cannot grasp or conceive
any principle of individuation apart from the fulness of the relations
to the surroundings.
But suppose now that Plato is talking about the higher man—the
four-dimensional being that is limited in our external experience to a
three-dimensional world. Do not his words begin to have a meaning? Such
a being would have a consciousness of motion which is not as the motion
he can see with the eyes of the body. He, in his own being, knows a
reality to which the outward matter of this too solid earth is flimsy
superficiality. He too knows a mode of being, the fulness of relations,
in which can only be represented in the limited world of sense, as the
painter unsubstantially portrays the depths of woodland, plains, and
air. Thinking of such a being in man, was not Plato’s line well divided?
It is noteworthy that, if Plato omitted his doctrine of the independent
origin of ideas, he would present exactly the four-dimensional
argument; a real thing as we think it is an idea. A plane being’s idea
of a square object is the idea of an abstraction, namely, a geometrical
square. Similarly our idea of a solid thing is an abstraction, for
in our idea there is not the four-dimensional thickness which is
necessary, however slight, to give reality. The argument would then
run, as a shadow is to a solid object, so is the solid object to the
reality. Thus A and B´ would be identified.
In the allegory which I have already alluded to, Plato in almost as
many words shows forth the relation between existence in a superficies
and in solid space. And he uses this relation to point to the
conditions of a higher being.
He imagines a number of men prisoners, chained so that they look at
the wall of a cavern in which they are confined, with their backs to
the road and the light. Over the road pass men and women, figures and
processions, but of all this pageant all that the prisoners behold
is the shadow of it on the wall whereon they gaze. Their own shadows
and the shadows of the things in the world are all that they see, and
identifying themselves with their shadows related as shadows to a world
of shadows, they live in a kind of dream.
Plato imagines one of their number to pass out from amongst them
into the real space world, and then returning to tell them of their
condition.
Here he presents most plainly the relation between existence in a plane
world and existence in a three-dimensional world. And he uses this
illustration as a type of the manner in which we are to proceed to a
higher state from the three-dimensional life we know.
It must have hung upon the weight of a shadow which path he
took!—whether the one we shall follow toward the higher solid and the
four-dimensional existence, or the one which makes ideas the higher
realities, and the direct perception of them the contact with the truer
world.
Passing on to Aristotle, we will touch on the points which most
immediately concern our enquiry.
Just as a scientific man of the present day in reviewing the
speculations of the ancient world would treat them with a curiosity
half amused but wholly respectful, asking of each and all wherein lay
their relation to fact, so Aristotle, in discussing the philosophy
of Greece as he found it, asks, above all other things: “Does this
represent the world? In this system is there an adequate presentation
of what is?”
He finds them all defective, some for the very reasons which we esteem
them most highly, as when he criticises the Atomic theory for its
reduction of all change to motion. But in the lofty march of his reason
he never loses sight of the whole; and that wherein our views differ
from his lies not so much in a superiority of our point of view, as
in the fact which he himself enunciates—that it is impossible for one
principle to be valid in all branches of enquiry. The conceptions
of one method of investigation are not those of another; and our
divergence lies in our exclusive attention to the conceptions useful
in one way of apprehending nature rather than in any possibility we
find in our theories of giving a view of the whole transcending that of
Aristotle.
He takes account of everything; he does not separate matter and the
manifestation of matter; he fires all together in a conception of a
vast world process in which everything takes part—the motion of a grain
of dust, the unfolding of a leaf, the ordered motion of the spheres in
heaven—all are parts of one whole which he will not separate into dead
matter and adventitious modifications.
And just as our theories, as representative of actuality, fall before
his unequalled grasp of fact, so the doctrine of ideas fell. It is
not an adequate account of existence, as Plato himself shows in his
“Parmenides”; it only explains things by putting their doubles beside
them.
For his own part Aristotle invented a great marching definition which,
with a kind of power of its own, cleaves its way through phenomena
to limiting conceptions on either hand, towards whose existence all
experience points.
In Aristotle’s definition of matter and form as the constituent of
reality, as in Plato’s mystical vision of the kingdom of ideas, the
existence of the higher dimensionality is implicitly involved.
Substance according to Aristotle is relative, not absolute. In
everything that is there is the matter of which it is composed, the
form which it exhibits; but these are indissolubly connected, and
neither can be thought without the other.
The blocks of stone out of which a house is built are the material for
the builder; but, as regards the quarrymen, they are the matter of the
rocks with the form he has imposed on them. Words are the final product
of the grammarian, but the mere matter of the orator or poet. The atom
is, with us, that out of which chemical substances are built up, but
looked at from another point of view is the result of complex processes.
Nowhere do we find finality. The matter in one sphere is the matter,
plus form, of another sphere of thought. Making an obvious application
to geometry, plane figures exist as the limitation of different
portions of the plane by one another. In the bounding lines the
separated matter of the plane shows its determination into form. And
as the plane is the matter relatively to determinations in the plane,
so the plane itself exists in virtue of the determination of space. A
plane is that wherein formless space has form superimposed on it, and
gives an actuality of real relations. We cannot refuse to carry this
process of reasoning a step farther back, and say that space itself is
that which gives form to higher space. As a line is the determination
of a plane, and a plane of a solid, so solid space itself is the
determination of a higher space.
As a line by itself is inconceivable without that plane which it
separates, so the plane is inconceivable without the solids which
it limits on either hand. And so space itself cannot be positively
defined. It is the negation of the possibility of movement in more than
three dimensions. The conception of space demands that of a higher
space. As a surface is thin and unsubstantial without the substance of
which it is the surface, so matter itself is thin without the higher
matter.
Just as Aristotle invented that algebraical method of representing
unknown quantities by mere symbols, not by lines necessarily
determinate in length as was the habit of the Greek geometers, and so
struck out the path towards those objectifications of thought which,
like independent machines for reasoning, supply the mathematician
with his analytical weapons, so in the formulation of the doctrine
of matter and form, of potentiality and actuality, of the relativity
of substance, he produced another kind of objectification of mind—a
definition which had a vital force and an activity of its own.
In none of his writings, as far as we know, did he carry it to its
legitimate conclusion on the side of matter, but in the direction of
the formal qualities he was led to his limiting conception of that
existence of pure form which lies beyond all known determination
of matter. The unmoved mover of all things is Aristotle’s highest
principle. Towards it, to partake of its perfection all things move.
The universe, according to Aristotle, is an active process—he does
not adopt the illogical conception that it was once set in motion
and has kept on ever since. There is room for activity, will,
self-determination, in Aristotle’s system, and for the contingent and
accidental as well. We do not follow him, because we are accustomed to
find in nature infinite series, and do not feel obliged to pass on to a
belief in the ultimate limits to which they seem to point.
But apart from the pushing to the limit, as a relative principle
this doctrine of Aristotle’s as to the relativity of substance is
irrefragible in its logic. He was the first to show the necessity
of that path of thought which when followed leads to a belief in a
four-dimensional space.
Antagonistic as he was to Plato in his conception of the practical
relation of reason to the world of phenomena, yet in one point he
coincided with him. And in this he showed the candour of his intellect.
He was more anxious to lose nothing than to explain everything. And
that wherein so many have detected an inconsistency, an inability to
free himself from the school of Plato, appears to us in connection with
our enquiry as an instance of the acuteness of his observation. For
beyond all knowledge given by the senses Aristotle held that there is
an active intelligence, a mind not the passive recipient of impressions
from without, but an active and originative being, capable of grasping
knowledge at first hand. In the active soul Aristotle recognised
something in man not produced by his physical surroundings, something
which creates, whose activity is a knowledge underived from sense.
This, he says, is the immortal and undying being in man.
Thus we see that Aristotle was not far from the recognition of the
four-dimensional existence, both without and within man, and the
process of adequately realising the higher dimensional figures to which
we shall come subsequently is a simple reduction to practice of his
hypothesis of a soul.
The next step in the unfolding of the drama of the recognition of
the soul as connected with our scientific conception of the world,
and, at the same time, the recognition of that higher of which a
three-dimensional world presents the superficial appearance, took place
many centuries later. If we pass over the intervening time without a
word it is because the soul was occupied with the assertion of itself
in other ways than that of knowledge. When it took up the task in
earnest of knowing this material world in which it found itself, and of
directing the course of inanimate nature, from that most objective aim
came, reflected back as from a mirror, its knowledge of itself.