Thus the memory of the definite properties of objects, which they have seen, helps animals to orientate in the world of phenomena. But, as a rule, when faced with new phenomena, animals are much more helpless than man.
Animals see two dimensions. They constantly sense the third dimension but do not see it. They sense it as something transient, as we sense time.
The surfaces which animals see possess for them many strange properties; these are, first of all numerous and varied movements.
It has been said already that all illusory movements must be perfectly real for them. These movements seem real to us also, but we know them to be illusory, as for instance the turning round of a house as we drive past, the springing up of a tree from round the corner, the movement of the moon among the clouds and so on.
In addition, many other movements will exist for animals, which we do not suspect. Actually a great many objects, completely motionless for us — indeed all objects — must appear to animals as moving. And it is precisely in these movements that the third dimension of solids will be manifested for them, i.e. The third dimension of solids will appear to them as motion.
Let us try to imagine how an animal perceives objects of the external world.
Let us suppose that a large disc is placed before an animal and, beside it, a large sphere of the same diameter.
Facing them directly at a certain distance, the animal will see two circles. If it starts walking round them, the animal will notice that the sphere remains a circle but the disc gradually narrows and becomes a narrow strip. As the animal continues to move round it, the strip begins to widen and gradually becomes again a circle. The sphere will not change its form as the animal moves round it, but strange phenomena will begin to occur in it as the animal draws near.
Let us try to understand how the animal will perceive the surface of the sphere as distinct from the surface of the disc.
One thing is certain — it will perceive a spherical surface differently from us. We perceive convexity or sphericity as a property common to many surfaces. Owing to the nature of its mental apparatus, the animal should perceive sphericity as an individual property of the given sphere. What should sphericity look like, taken as an individual property of a given sphere?
We can say with the utmost conviction that sphericity will appear to the animal as a movement of the surface it sees.
When the animal comes near to the sphere, in all probability what happens is something like this: the surface the animal sees springs into rapid motion; its centre projects forward, and all the other points begin to recede from the centre with a velocity proportionate to their distance from the centre (or the square of their distance from the centre).
This is the way in which the animal must sense a spherical surface. It is reminiscent of the way we sense sound. At a certain distance from the sphere the animal sees it as a plane. Approaching it and touching some point of the sphere, it sees that the relation of all the other points to that point has changed as compared with what it should be on a plane, as if all the other points have moved, have drawn aside. Touching another point it again sees all the other points withdrawing from it.
This property of the sphere will appear as its motion, as ‘vibration’. And indeed the sphere will resemble a vibrating, undulating surface. In the same way any angle of a motionless object must appear as motion to the animal.
The animal can see an angle of a three-dimensional object only if it moves past it, and in that case the object will seem to have turned — a new side has appeared, and the old side has receded or moved aside. An angle will be perceived as a turning, a movement of the object, i.e. as something transient, temporal, i.e. as a change in the state of the object. Remembering the angles met with before — which the animal has seen as the motion of bodies — it will regard them as gone, finished, vanished, belonging to the past.
Of course, the animal cannot reason thus, but it will act as though this was its reasoning.
If the animal could think of phenomena (i.e. angles and curved surfaces), which have not yet entered its life, it would no doubt represent them to itself only in time. In other words, the animal could not allow them any real existence at the present moment when they have not yet appeared. If it could express an opinion about them, it would say that these angles exist as a potentiality, that they will be, but that at present they are not.
For a horse, the corner of a house past which it runs every day, is a phenomenon, which recurs in certain circumstances, but which still takes place only in time; it is not a spatial and constant property of the house.
For the animal an angle must be a time-phenomenon, instead of being a space-phenomenon as it is for us.
Thus we see that the animal will perceive the properties of our third dimension as movements and will refer these properties to time, to the past or future, or to the present, i.e. to the moment of transition of the future into the past.
This is an extremely important point and contains the key to the understanding of our own perception of the world; consequently we must examine it in greater detail.
So far we have considered higher animals: a dog, a cat, a horse. Let us now take a lower animal — a snail for example. We know nothing about its inner life, but we may be sure that its perception is very different from ours. In all probability a snail’s sensations of its surroundings are very vague. It probably feels warmth, cold, light, darkness, hunger, and instinctively (i.e. incited by the pleasure-pain guidance) it crawls towards the uneaten edge of the leaf it sits on and draws away from a dead leaf. Its movements are governed by pleasure-pain; it always advances towards the one and retreats from the other. It always moves on one line — from the unpleasant towards the pleasant. And, in all probability, it knows and senses nothing except this line. This line constitutes the whole of its world. All the sensations entering from outside are sensed by the snail on its line of motion. And these come to it out of time — from potentiality they become actuality. For a snail the whole of our universe exists in the future and the past, i.e. in time. Only one line exists in the present; all the rest lies in time. It is more than probable that a snail is not aware of its own movements; making efforts with its whole body it moves forward towards the fresh edge of the leaf, but it seems to it that the leaf moves towards it, coming into being at that moment, appearing out of time, as the morning appears to us.
A snail is a one-dimensional being.
Higher animals — a dog, a cat, and a horse — are two-dimensional beings. To them space appears as a surface, a plane. Everything outside this plane lies for them in time.
Thus we see that a higher animal — a two-dimensional being as compared to a one-dimensional — extracts one more dimension out of time.
The world of a snail has one dimension — our second and third dimensions lie for it in time.
The world of a dog has two dimensions — our third dimension lies for it in time.
An animal may remember all the ‘phenomena’ it has observed, i.e. all the properties of three-dimensional bodies it has come into contact with, but it cannot know that that which for it is a recurring phenomenon is in reality a permanent property of a three-dimensional body — an angle, or curvature, or convexity.
This is the psychology of the perception of the world by a two-dimensional being.
For it a new sun will rise every day. Yesterday’s sun has gone and will never recur again. Tomorrow’s sun does not yet exist.
Rostand failed to understand the psychology of Chantecler. The cock could not think that he awakened the sun by his crowing. For him the sun does not go to sleep — it recedes into the past, vanishes, is annihilated, and ceases to be. Tomorrow, if it comes, there will be a new sun; just as for us there is a new spring each year. In order to be the sun cannot wake up; it must come into being, be born. An animal (if it could think without losing its characteristic psychology) could not believe in the appearance today of the same sun that was there yesterday. This is human reasoning.
For an animal a new sun rises every morning, just as for us a new morning comes every day, a new spring every year.
An animal is incapable of understanding that the sun is one and the same, whether today or yesterday — exactly as we probably cannot understand that the morning is one, and the spring is one.
The motion of objects which, for us, is not illusory but real, such as the motion of a rotating wheel or a moving carriage and so on, must, for an animal, differ greatly from the motion it sees in all objects which are motionless for us — that motion in the guise of which it sees the third dimension of bodies. This first motion (i.e. motion which is also real for us) must appear to it spontaneous, alive.
And these two kinds of motion will be incommensurable for it. An animal will be able to measure an angle or a convex surface, although it will not understand its true meaning and will regard it as motion. But it will never be able to measure real motion, i.e. motion that is real for us. To do this it is necessary to have our conception of time and measure all movements in relation to some more constant motion, i.e. compare all movements with one. As an animal has no concepts, it will not be able to do this. Therefore, movements of objects which are real for us will be incapable of measurement, and thus incommensurable with other movements which, for it, are real and capable of measurement, but for us are illusory, constituting in reality the third dimension of bodies.
The latter is inevitable. If an animal senses and measures as motion that which is not motion, it is clear that it cannot apply the same measure to that which is and that which is not motion.
But this does not mean that an animal cannot know the character of movements proceeding in our world and conform to them. On the contrary, we see that an animal orientates perfectly among the movements of objects of our three-dimensional world. In this it is helped by instinct, i.e. capacity, evolved through hundreds of centuries of selection, of performing expedient actions without consciousness of purpose. And an animal discriminates perfectly well between movements happening round it.
But, distinguishing between two kinds of phenomena — two kinds of motion — an animal is bound to explain one of them by some inner inexplicable property of objects, i.e. it will probably regard that kind of motion as the result of the animation of objects, and will regard moving objects as alive.
A kitten plays with a ball or with its own tail because the ball or the tail runs away from it.
A bear will fight with a beam until the beam throws him off the tree, because in the swinging beam he feels something alive and hostile.
A horse shies from a bush because the bush has suddenly turned round and waved a branch.
In the latter case the bush may not have moved at all — it was the horse that was running. But it appeared to move; therefore, it was alive. Probably everything that moves is alive for an animal. Why does a dog bark so furiously at a passing carriage? We do not quite understand it. We do not see how a passing carriage turns, twists and grimaces in the eyes of a dog. It is full of life — the wheels, the roof, the mudguards, the seats, and the passengers — all this is moving, turning…
Now let us summarize our deductions.
We have established that a man possesses sensations, representations and concepts; that higher animals possess sensations and representations, and lower animals only sensations. We deduced that an animal has no concepts mainly from the fact that it has no words, no speech. We have further established that, having no concepts, animals cannot comprehend the third dimension and only see the world as a surface. In other words they have no means, no instrument, for correcting their wrong sensations of the world. Then we found that, seeing the world as a surface, animals see on this surface a great many movements non-existent for us. That is, all those properties of bodies which we regard as the properties of their three-dimensionality, must appear as movements to them. Thus an angle and a spherical surface must appear to them as motion of the plane. Further, we came to the conclusion that everything that, for us, belongs to the domain of the third dimension as something constant, animals must regard as transient occurrences happening to objects — as time-phenomena.
Thus, in all its relations to the world an animal proves to be completely analogous to the unreal two-dimensional being, which we have supposed, lived on a plane. The whole of our world appears to an animal as a plane through which phenomena are passing, moving according to time or in time.
So we can say that we have established the following: that with a certain limitation of the mental apparatus which perceives the external world, for a subject possessing such an apparatus the whole aspect and all the properties of the world must change. And two subjects, living side by side but possessing different mental apparatuses, must live in different worlds — the properties of the extension of the world must be quite different for them. Moreover, we have seen conditions — not artificial and invented but actually existing in nature, i.e. the mental conditions of the life of animals — in which the world appears as a plane or even as a line.
In other words we have established that the three-dimensional extension of the world depends for us on the properties of our mental apparatus; or, that the world’s three-dimensionality is not its own property, but merely the property of our perception of the world.
To put it differently, the three-dimensionality of the world is the property of its reflection in our consciousness.
If all this is so, it is clear that we have really proved the dependence of space on space-sense. And, since we have proved the existence of a space-sense lower than ours, by this very fact we have proved the possibility of a space-sense higher than ours.
And we must admit that if a fourth unit of thinking becomes formed in us, as different from the concept as the concept is different from the representation, then, simultaneously with this, there will appear for us in the surrounding world a fourth characteristic which we may call geometrically a fourth direction or a fourth perpendicular, because this characteristic will contain properties of objects perpendicular to all properties known to us and not parallel to any of them. In other words, we shall see or feel ourselves not in a space of three, but of four dimensions, and the surrounding objects as well as our own bodies will reveal the general properties of the fourth dimension which we had not noticed before or which we had regarded as individual properties of objects (or their motion), just as animals regard the extension of objects in the third dimension as their motion.
Having seen or felt ourselves in the world of four dimensions, we shall find that the world of three dimensions has not and never had any real existence that it was a creation of our fantasy, a phantom, a spectre, a delusion, an optical illusion, anything you like, but not reality.
All this is far from being a ‘hypothesis’, a supposition; it is an exact fact, as much of a fact as the existence of infinity. For the sake of its own existence, positivism had somehow to do away with infinity or at least to call it a ‘hypothesis’, which may or may not be true. But infinity is not a hypothesis; it is a fact. And just such a fact is also the multi-dimensionality of space and all that it implies, i.e. the unreality of everything three-dimensional.
