The receptivity of the world by a man and by an animal. Illusions of the animal and its lack of control of the receptive faculties. The world of moving planes. Angles and curves considered as motion. The third dimension as motion. The animal's two-dimensional view of our three-dimensional world. The animal as a real two-dimensional being. Lower animals as one-dimensional beings. The time and space of a snail. The time-sense as an imperfect space-sense. The time and space of a dog. The change in the world coincident with a change in the psychic apparatus. The proof of Kant's problem. The three-dimensional world—an illusionary perception.
WE have established the enormous difference existing between the psychology of a man and of an animal. This difference undoubtedly profoundly affects the receptivity of the outer world by the animal. But how and in what? This is exactly what we do not know, and what we shall try to discover.
To this end we shall return to our receptivity of the world, investigate in detail the nature of that receptivity, and then imagine how the animal, with its more limited psychic equipment, receives its impression of the world.
Let us note first of all that we receive the most incorrect impressions of the world as regards its outer form and aspect. We know that the world consists of solids, but we see and touch only surfaces. We never see and touch a solid. The solid—this is indeed a concept, composed of a series of perceptions, the result of reasoning and experience. For immediate sensation, surfaces alone exist. Sensations of gravity, mass, volume, which we mentally associate with the "solid," are in reality associated with the sensations of surfaces. We only know that the sensation comes from the solid, but the solid itself we never sense. Perhaps it would be possible to call the complex sensation of surfaces: weight, mass, density, resistance, "the sensation of a solid," but rather do we combine mentally all these sensations into one, and call that composite sensation a solid. We sense directly only surfaces; the weight and resistance of the solid, as such, we never separately sense.
But we know that the world does not consist of surfaces: we know that we see the world incorrectly, and that we never see it as it is, not alone in the philosophical meaning of the expression, but in the most simple geometrical meaning. We have never seen a cube, a sphere, etc., but only their surfaces. Knowing this, we mentally correct that which we see. Behind the surfaces we think the solid. But we can never even represent the solid to ourselves. We cannot imagine the cube or the sphere seen, not in perspective, but simultaneously from all sides.
It is clear that the world does not exist in perspective; nevertheless we cannot see it otherwise. We see everything only in perspective; that is, in the very act of receptivity the world is distorted in our eye, and we know that it is distorted. We know that it is not such as it appears, and mentally we are continuously correcting that which the eye sees, substituting the real content for those symbols of things which sight reveals.
Our sight is a complex faculty. It consists of visual sensations plus the memory of sensations of touch. The child tries to feel with its finger-tips everything that it sees—the nose of its nurse, the moon, the reflection of sun rays from the mirror on the wall. Only gradually does it learn to discern the near and the distant by means of sight alone. But we know that even in mature age we are easily subject to optical illusions.
We see distant objects as flat, even more incorrectly, because relief is after all a symbol revealing a certain property of objects. A man at a long distance is pictured to us in silhouette. This happens because we never feel anything at a long distance, and the eye has not been taught to discern the difference in surfaces which at short distances are felt by the finger-tips.
We can never see, even in the minute, any part of the outer world as it is, that is, as we know it. We can never see the desk or the wardrobe all at once, from all sides and inside. Our eye distorts the outside world in a certain way, in order that, looking about, we may be able to define the position of objects relatively to ourselves. But to look at the world from any other standpoint than our own is impossible for us, nor can we ever see it correctly, without distortion by our sight.
Relief and perspective—these constitute the distortions of the object by our eye. They are optical illusions, delusions of sight. The cube in perspective is but a conventional sign of the three-dimensional cube, and all that we see is the conditional image of that conditionally real three-dimensional world with which our geometry deals, and not that world itself. On the basis of what we see we surmise that it exists in reality. We know that what we see is incorrect, and we think of the world as other than it appears. If we had no doubt about the correctness of our sight, if we knew that the world were such as it appears, then obviously we should think of the world in the manner in which we see it. In reality we are constantly engaged in making corrections.
It is clear that the ability to make corrections in that which the eye sees demands, undoubtedly, the possession of the concept, because the corrections are made by a process of reasoning, which is impossible without concepts. Deprived of the faculty to make corrections in that which the eye sees we should have a different outlook on the world, i.e., much of that which is we should see incorrectly; we should not see much of that which is, but we should see much of that which does not exist in reality at all. First of all, we should see an enormous number of non-existent motions. Every motion of ours in our direct sensation of it, is bound up with the motion of everything around us. We know that this motion is an illusory one, but we see it as real. Objects turn in front of us, run past us, overtake one another. If we are riding slowly past houses, these turn slowly, if we are riding fast they turn quickly; also, trees grow up before us unexpectedly, run away and disappear.
This seeming animation of objects, coupled with dreams, has always inspired, and still inspires the fairy tale.
The "motions" of objects, to a person in motion, are very complex indeed. Observe how strangely the field of wheat behaves just beyond the window of the car in which you are riding. It runs to the very window, stops, turns slowly around itself and runs away. The trees of the forest run apparently at different speeds, overtaking one another. The entire landscape is one of illusory motion. Behold also the sun, which even up to the present time "rises" and "sets" in all languages—this "motion" having been in the past so passionately defended!
This is all seeming, and though we know that these motions are illusory, we see them nevertheless, and sometimes we are deluded. To how many more illusions should we be subject had we not the power of mentally analyzing their determining causes, but were obliged to believe that everything exists as it appears!
I see it; therefore this exists.
This affirmation is the principal source of all illusions. To be true, it is necessary to say:
I see it; therefore this does not exist—or at least, I see it; therefore this is not so.
Although we can say the last, the animal cannot, for to its apprehension things are as they appear. It must believe what it sees.
How does the world appear to the animal?
The world appears to it as a series of complicated moving surfaces. The animal lives in a world of two dimensions. Its universe has for it the properties and appearance of a surface. And upon this surface transpire an enormous number of different movements of a most fantastic character.
Why should the world appear to the animal as a surface?
First of all, because it appears as a surface to us.
But we know that the world is not a surface, and the animal can, not know it. It accepts everything just as it appears. It is powerless to correct the testimony of its eyes—or it cannot do so to the extent that we do.
We are able to measure in three mutually independent directions: the nature of our mind permits us to do this. The animal can measure simultaneously in two directions only—it can never measure in three directions at once. This is due to the fact that, not possessing concepts, it is unable to retain in the mind the idea of the first two directions, for measuring the third.
Let me explain this more exactly.
Suppose we imagine that we are measuring the cube.
In order to measure the cube in three directions, it is necessary while measuring in one direction, to keep in mind two others—to remember. But it is possible to keep them in mind as concepts only, that is, associating them with different concepts—pasting upon them different labels. So, pasting upon the first two directions the labels of length and breadth, it is possible to measure the height. It is impossible otherwise. As perceptions, the first two measurements of the cube are completely identical, and assuredly will mingle into one in the mind. The animal, without the aid of concepts, cannot paste upon the first two measurements the labels of length and breadth. Therefore, at the moment when it begins to measure the height of the cube, the first two measurements will be confused in one. The animal, attempting to measure the cube by means of perceptions only without the aid of concepts, will be like a cat I once observed. Her kittens—five or six in number—she dragged asunder into different rooms, and could not then collect them together. She seized one, put it beside another, ran for a third and brought it to the first two, but then she seized the first and carried it away to another room, putting it beside the fourth; after that she ran back, seized the second and dragged it to the room containing the fifth, and so on. For a whole hour the cat had no rest with her kittens, she suffered severely, and could accomplish nothing. It is clear that she lacked the concepts which would enable her to remember how many kittens she had altogether.
It is in the highest degree important to understand the relation of the animal consciousness to the measuring of bodies.
The great point is that the animal sees surfaces only. (We may say this with complete assurance, because we ourselves see surfaces only.) Thus seeing only surfaces the animal can imagine but two dimensions. The third-dimension, in contradistinction to the other two, can only be thought; that is, this dimension must be a concept; but animals do not possess concepts. The third dimension like the others appears as a perception. Therefore, at the moment of its appearance, the first two will inevitably mingle into one. The animal is capable of perceiving the difference between two dimensions: the difference between three it cannot perceive. This difference must be known beforehand, and to know it concepts are necessary.
Identical perceptions mix into one for the animal, just as we ourselves confuse two simultaneous, similar phenomena proceeding from the same point. For the animal it will be one phenomenon, just as for us all similar, simultaneous phenomena proceeding from a single point will be one phenomenon.
Therefore the animal will see the world as a surface, and will measure this surface in two directions only.
But how is it possible to explain the fact that the animal, inhabiting a two-dimensional world, or rather, perceiving itself as in a two-dimensional world, is perfectly oriented in our three-dimensional world? How explain the fact that the bird flies up and down, sideways and straight ahead—in all three directions; that the horse jumps over ditches and barriers; that the dog and cat appear to, understand the properties of depth and height simultaneously with those of length and breadth?
In order to explain these things it is necessary to return to the fundamental principles of animal psychology. It has been previously shown that many properties of objects remembered by us as general properties of genus, class, species, are remembered by animals as individual properties of objects. To orientate in this enormous reserve of individual properties preserved in the memory, animals are assisted by the emotional tone which is linked up in them with each perception and each remembered sensation.
For example, an animal knows two roads as two entirely separate phenomena having nothing in common; that is, one road consists of a series of definite perceptions colored by definite emotional tones; the other phenomenon—the other road—consists of another series of definite perceptions colored with other tones. We say that this, that, and the other are roads. One leads to one place, a second to another. For an animal the two roads have nothing in common. But it remembers in their proper sequence all the emotional tones which are linked with the first road and with the second one, and it therefore remembers both roads with their turns, ditches, fences, etc.
Thus the remembering of definite properties of observed objects helps the animal to find itself in the world of phenomena. But as a rule before new phenomena an animal is much more helpless than a man.
An animal sees two dimensions; the third dimension it senses constantly, but does not see. It senses the third dimension as something transient, just as we sense time.
The surfaces which an animal sees possess for it many strange properties; first of all, numerous and various motions.
As has been said already, all those illusory motions which seem to us real, but which we know to be illusory, are entirely real to the animal: the turning about of the houses as we ride past, the growth of a tree out of some corner, the passing of the moon between clouds, etc., etc.
But in addition to all this, many motions must exist for the animal of which we have no suspicion. The fact is that innumerable objects quite immobile for us—properly all objects—must seem to the animal to be in motion; AND THE THIRD DIMENSION OF SOLIDS WILL APPEAR TO IT IN THESE MOTIONS; i.e., THE THIRD DIMENSION OF SOLIDS WILL APPEAR TO IT AS A MOTION.
Let us try to imagine how the animal perceives the objects of the outer world.
Suppose it is confronted with a large disc, and simultaneously with a large sphere of the same diameter.
Standing directly opposite them at a certain distance, the animal will see two circles. Beginning to walk around them, it will observe that the sphere remains a circle, while the disc gradually narrows, transforming itself into a narrow strip. On moving farther around, the strip begins to expand and gradually transforms itself into a circle. The sphere will not change during this circumambulation. But when the animal approaches toward it certain strange phenomena ensue.
Let us try to understand how the animal will perceive the surface of the sphere as contrasted with the surface of the disc.
One thing is sure: it will perceive the spherical surface differently from us. We perceive convexity or sphericality as a common property of many surfaces. The animal, on the contrary, because of the very properties of its psychic apparatus, will perceive that sphericality as an individual property of a given sphere. Now how will this sphericality as an individual property of a given sphere appear to it?
We may declare with complete assurance that the sphericality will appear to the animal as a movement on the surface which it sees.
During the approach of the animal toward the sphere something like the following must happen: the surface which the animal sees starts to move quickly; its center spreads out, and all of the other points run away from the center with a velocity proportional to their distance from the center (or the square of their distance from the center).
It is in this way that the animal senses the spherical surface—much as we sense sound.
At a certain distance from the sphere the animal perceives it as a plane. Approaching or touching some point on the sphere it sees that all other points have changed with relation to this particular point, they have all altered their position on the plane—have moved to one side, as it were. Touching another point, it sees that all the rest have moved in similar fashion.
This property of the sphere will appear as its motion, its "vibration." The sphere will actually resemble a vibrating, oscillating surface, in the same way that each angle of an immobile object will appear to the animal as a motion.
The animal can see an angle of a three-dimensional object only while moving past it, and during the time it takes, the object will seem to the animal to have turned—a new side has appeared, and the side first seen has disappeared or moved away. The angle will be perceived as rotation, as the motion of the object, i.e., as something transient, temporal, as a change of state in the object. Remembering the angles which it has seen before—seen as the motion of bodies—the animal will consider that they have ceased, have ended, have disappeared—that they are in the past.
Of course the animal cannot reason in this way, but it acts as though it had thus reasoned.
Could the animal think about those phenomena which have not yet entered into its life (i.e., angles and curved surfaces) it would undoubtedly imagine them in time only: it could not prefigure for them any real existence at the present moment when they have not yet appeared. And were it able to express an opinion on this subject, it would say that angles exist in potentiality, that they will be, but that for the present they do not exist.
The angle of a house past which a horse runs every day is a phenomenon, repeating under certain circumstances, but nevertheless a phenomenon proceeding in time, and not a spatial and constant property of the house.
For the animal the angle will be a temporal phenomenon and not a spatial one, as it is for us.
Thus we see that the animal will perceive the properties of our third dimension as motions, and will refer these properties to time, i.e., to the past or future, or to the present—the moment of the transition of the future into the past.
This circumstance is in the highest degree important, for there-in lies the key to our own receptivity of the world; we shall therefore examine into it more in detail.
Up to the present time we have taken into consideration only the higher animals: the dog, the cat, the horse. Let us now try the lower: let us take the snail. We know nothing about its inner life, but undoubtedly its receptivity resembles ours scarcely at all. In all probability the snail possesses some obscure sensations of its environment. Probably it feels heat, cold, light, darkness, hunger—and it instinctively (i.e., urged by pleasure-pain guidance) strives to reach the uneaten edge of the leaf on which it rests, and instinctively avoids the dead leaf. Its movements are guided by pleasure-pain: it constantly strives toward the one, and away from the other. It always moves upon a single line, from the unpleasant to the pleasant, and in all probability except for this line it is not conscious of anything and does not sense anything. This line is its entire world. All sensations, entering from the outside, the snail senses upon this line of its motion, and these come to it out of time—from the potential they become the present. For the snail our entire universe exists in the future and in the past—i.e., in time. In space only one line exists; all the rest is time. It is more than probable that the snail is not conscious of its movements. Making efforts with its entire body it moves forward to the fresh edge of the leaf, but it seems as if the leaf were coming to it, appearing at that moment, coming out of time as the morning comes to us.
The snail is a one-dimensional being.
The higher animals—the dog, cat, the horse—are two-dimensional beings. To the higher animal all space appears as a surface, as a plane. Everything out of this plane lives for it in time.
Thus we see that the higher animal—the two-dimensional being compared with the one-dimensional—extracts or captures from time one more dimension.
The world of a snail has one dimension; our second and third dimensions are for it in time.
The world of a dog is two-dimensional; our third dimension is for it in time.
An animal can remember all "phenomena" which it has observed, i.e., all properties of three-dimensional solids with which it has come in contact, but it cannot know that the (for it) recurring phenomenon is a constant property of the three-dimensional solid—an angle, curvature, or convexity.
Such is the psychology of the receptivity of the world by a two-dimensional being.
For such a being a new sun will rise every day. Yesterday's sun is gone, and will not appear again; tomorrow's does not as yet exist.
Rostand did not understand the psychology of "Chantecler." The cock could not think that he woke up the sun by his crowing. To him the sun does not go to sleep, it goes into the past, disappears, suffers annihilation, ceases to be. If it comes on the morrow it will be a new sun, just as for us with every new year comes a new spring. In order to be the sun shall not wake up, but arise, be born. The cock (if it could think without losing its characteristic psychology) could not believe in the appearance today of the same sun which was yesterday. This is purely human reasoning.
For the animal a new sun rises every morning, just as for us a new morning comes with every day and a new spring with every year.
The animal is not in a position to understand that the sun is the same yesterday and today, EXACTLY IN THE SAME WAY THAT WE PROBABLY CANNOT UNDERSTAND THAT THE MORNING IS THE SAME AND THE SPRING IS THE SAME.
The motion of objects which is not illusory, even for us, but a real motion, like that of a revolving wheel, a passing carriage, and so on, will differ for the animal very much from that motion which it sees in all objects which are for us immobile—i.e., from that motion in which the third dimension of solids is as it were revealed to it. The first mentioned motion (real for us) will seem to the animal arbitrary, alive.
And these two kinds of motion will be incommensurable for it.
The animal will be in a position to measure an angle or a convex surface, though not understanding their true nature, and though regarding them as motion. But true motion, i.e., that which is true motion to us, it will never be in a position to measure, because for this it is necessary to possess our concept of time, and to measure all motions with reference to some one more constant motion, i.e., to compare all motions with some one. Without concepts the animal is powerless to do this. Therefore the (for us) real motions of objects will be incommensurable for it, and being incommensurable, will be incommensurable with other motions which are real and measurable for it, but which are illusory for us—motions which in reality represent the third dimension of solids.
This last conclusion is inevitable. If the animal apprehends and measures as motion that which is not motion, clearly it cannot measure by one and the same standard that which is motion and that which is not motion.
But this does not mean that it cannot know the character of motions going on in the world and cannot conform itself to them. On the contrary, we see that the animal orientates itself perfectly among the motions of the objects of our three-dimensional world. Here comes into play the aid of instinct, i.e., the ability, developed by millenniums of selection, to act expediently without consciousness of purpose. Moreover, the animal discerns perfectly the motions going on around it.
But discerning two kinds of phenomena, two kinds of motion, the animal will explain one of them by means of some incomprehensible inner property of objects, i.e., in all probability it will regard this motion as the result of the animation of objects, and the moving objects as animated beings.
The kitten plays with the ball or with its tail because ball and tail are running away from it.
The bear will fight with the beam which threatens to throw him off the tree, because in the swinging beam he senses something alive and hostile.
The horse is frightened by the bush because the hush unexpectedly turned and waved a branch.
In the last case the bush need not even have moved at all, for the horse was running, and it seemed therefore as though the bush moved, and consequently that it was animated. In all probability all movement is thus animated for the animal. Why does the dog bark so desperately at the passing carriage? This is not entirely clear to us for we do not realize that to the eyes of the dog the carriage is turning, twisting, grimacing all over. It is alive in every part—the wheels, the top, the mud-guards, seats, passengers—all these are moving, turning.
Now let us draw certain conclusions from all of the foregoing.
We have established the fact that man possesses sensations, perceptions and concepts; that the higher animals possess sensations and perceptions, and the lower animals sensations only. The conclusion that animals have no concepts we deduced from the fact that they have no speech. Next we have established that having no concepts, animals cannot comprehend the third dimension, but see the world as a surface; i.e., they have no means—no instrument—for the correction of their incorrect sensations of the world. Furthermore, we have found that seeing the world as a surface, animals see upon this surface many motions which for us are non-existent. That is, all those properties of solids which we regard as the properties of three-dimensionality, animals represent to themselves as motions. Thus the angle and the spherical surface appear to them as the movements of a plane. After that we came to the conclusion that everything which we regard as constant in the region of the third dimension, animals regard as transient things which happen to objects—temporal phenomena.
Thus in all its relations to the world the animal is quite analogous to the imagined, unreal two-dimensional being living upon a plane. All our world appears to the animal as the plane through which phenomena are passing, moving upon time, or in time.
And so we may say that we have established the following: that under certain limitations of the psychic apparatus for receiving the outer world, for the subject possessing this apparatus, the entire aspect and all properties of the world will suffer change. And two subjects, living side by side, but possessing different psychic apparatus, will inhabit different worlds—the properties of the extension of the world will be different for them. And we observed the conditions, not invented for the purpose, not concocted in imagination, but really existing in nature; that is, the psychic conditions governing the lives of animals, under which the world appears as a plane or as a line.
That is to say, we have established that the three-dimensional extension of the world depends upon the properties of our psychic apparatus.
Or, that the three-dimensionality of the world is not its property, but a property of our receptivity of the world.
In other words, the three-dimensionality of the world is a property of its reflection in our consciousness.
If all this is so, then it is obvious that we have really proved the dependence of space upon the space-sense. And if we have proved the existence of a space-sense lower in comparison with ours, by this we have proved the possibility of a space-sense higher in comparison with ours.
And we shall grant that if in us there develops the fourth unit of reasoning as different from the concept as the concept is different from perception, so simultaneously with it will appear for us in the surrounding world a fourth characteristic which we may designate geometrically as the fourth direction or the fourth perpendicular, because in this characteristic will be included the properties of objects perpendicular to all properties known to us, and not parallel to any of them. In other words, we shall see, or we shall feel ourselves in a space not of three, but of four dimensions; and in the objects surrounding us, and in our own bodies, will appear common properties of the fourth dimension which we did not notice before, or which we regarded as individual properties of objects (or their motion), just as animals regard the extension of objects in the third dimension as their motion.
And when we shall see or feel ourselves in the world of four dimensions we shall see that the world of three dimensions does not really exist and has never existed: that it was the creation of our own fantasy, a phantom host, an optical illusion, a delusion—anything one pleases excepting only reality.
And all this is not an "hypothesis," not a supposition, but exact fact, just such a fact as the existence of infinity. For positivism to insure its existence it was necessary to annihilate infinity somehow, or at least to call it an "hypothesis" which may or may not be true. Infinity however is not an hypothesis, but a fact, and such a fact is the multi-dimensionality of space and all that it implies, namely, the unreality of everything three-dimensional.
99:1 In this connection, there have been some interesting observations made upon the blind who are just beginning to see.
In the magazine Slepetz (The Blind, 1912) there is a description from direct observation of how those born blind learn to see after the operation which restored their sight.
This is how a seventeen-year-old youth, who recovered his sight after the removal of a cataract, describes his impressions. On the third day after the operation he was asked what he saw. He answered that he saw an enormous field of light and misty objects moving upon it. These objects he did not discern. Only after four days did he begin to discern them, and after an interval of two weeks, when his eyes were accustomed to the light, he started to use his sight practically, for the discernment of objects. He was shown all the colors of the spectrum and he learned to distinguish them very soon, except yellow and green, which he confused for a long time. The cube, sphere and pyramid, p. 100 when placed before him seemed to him like the square, the flat disc, and the triangle. When the flat disc was put alongside the sphere he distinguished no difference between them. When asked what impression both kinds of figures produced on him just at first, he said that he noticed at once the difference between the cube and the sphere, and understood that they were not drawings, but was unable to deduce from them their relation to the square and to the circle, until he felt in his finger tips the desire to touch these objects. When he was allowed to take the cube, sphere and pyramid in his hands he at once identified these solids by the sense of touch, and wondered very much that he was unable to recognize them by sight. He lacked the perception of space, perspective. All objects seemed flat to him: though he knew that the nose protrudes, and that the eyes are located in cavities, the human face seemed flat to him. He was delighted with his recovered vision, but in the beginning it fatigued him to exercise it: the impressions oppressed and exhausted him. For this reason. though possessing perfect sight, he sometimes turned to the sense of touch as to repose.
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