The Fourth Dimension

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.

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.

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:

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.

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.

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.

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.

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.

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.

Then as A is to B, so is A¹ to B¹

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.