A new view of the Kantian problem. The books of Hinton. The "space-sense" and its evolution. A system for the development of a sense of the fourth dimension by exercises with colored cubes. The geometrical conception of space. Three perpendiculars—why three? Can everything existing be measured by three perpendiculars? The indices of existence. Reality of ideas. Insufficient evidence of the existence of matter and motion. Matter and motion are only logical concepts, like "good" and "evil."
AS already stated, Kant propounded the problem, but gave no solution of it, nor did he point the way to a solution. And not one of the known commentators, interpreters, followers or adversaries of Kant has found a solution, or the way to it.
I find the first flashes of a right understanding of the Kantian problem, and the first suggestions in regard to a possible way toward its solution, in the attempts at a new treatment of the problem of space and time, involving the concept of the "fourth dimension" and higher dimensions in general. An interesting synopsis of many things developed in this direction is that of C. H. Hinton, author of the books, A New Era of Thought, and The Fourth Dimension.
Hinton notes, among other things, that in commenting upon Kantian ideas, only their negative side is usually insisted upon, namely, the fact that we can cognize things in a sensuous way, in terms of space and time only, is regarded as an obstacle, hindering us from seeing what things in themselves really are, preventing the possibility of cognizing them as they are, imposing upon them that which is not inherent in them, shutting them off from us.
But [says Hinton] if we take Kant's statement simply as it is—not seeing in the spatial conception a hindrance to right receptivity—that we apprehend things by means of space—then it is equally allowable to consider our space sense not as a negative condition, hindering our perception of the world, but as a positive means by which the mind grasps its experiences, i.e., by which we cognize the world.
There is, in so many books in which the subject is treated, a certain air of despondency—as if this space apprehension were a kind of veil which shut us off from nature. But there is no need to adopt this feeling. The first postulate of this book is a full recognition of the fact that it is by means of space that we apprehend what is.
Space is the instrument of the mind.
Very often a statement which seems to be most deep and abstruse and hard to grasp, is simply the form into which deep thinkers have thrown a very simple and practical observation. And for the present let us look on Kant's great doctrine of space from a practical point of view, and it comes to this—it is important to develop the space sense, for it is the means by which we think about real things.
Now according to Kant [Hinton goes on to say] the space sense, or the intuition of space, is the most fundamental power of the mind. But I do not find anywhere a systematic and thorough-going education of the space sense. It is left to be organized by accident. Yet the special development of the space sense makes us acquainted with a whole series of new conceptions.
Fichte, Schelling, Hegel, have developed certain tendencies and have written remarkable books, but the true successors of Kant are Gauss and Lobachevsky.
For if our intuition of space is the means whereby we apprehend, then it follows that there may be different kinds of intuitions of space. Who can tell what the absolute space intuition is? This intuition of space must be colored, so to speak, by the conditions (of psychical activity) of the being which uses it.
By a remarkable analysis the great geometers above mentioned have shown that space is not limited as ordinary experience would seem to inform us, but that we are quite capable of conceiving different kinds of space.
(A New Era of Thought.)
Hinton invented a complicated system for the education and development of the space sense by means of exercises with groups the cubes of different colors. The books above mentioned are devoted to the exposition of this system. In my opinion Hinton's exercises are interesting from a theoretical standpoint, but they are practically valuable only for such as have the same turn of mind as Hinton's own.
Exercises of the mind according to his system must first of all lead to the development of the ability to imagine objects, not as the eye sees them, i.e., in perspective, but as they are geometrically—to learn to imagine the cube, for example, simultaneously from all sides. Moreover such a development of the imagination as overcomes the illusions of perspective results in the expansion of the limits of consciousness, thus creating new conceptions and augmenting the faculty for perceiving analogies.
Kant established the fact that the development of knowledge under the existing conditions of receptivity will not bring us any closer to things in themselves. But there are theories asserting that it is possible, if desired, to change the very conditions of receptivity, and thus to approach the true substance of things. In the books above referred to, Hinton tries to unite the scientific foundations of such theories.
Our space as we ordinarily think of it is conceived as limited—not in extent, but in a certain way which can only be realized when we think of our ways of measuring space objects. It is found that there are only three independent directions in which a body can be measured—it must have height, length and breadth, but it has no more than these dimensions, if any other measurement be taken in it, this new measurement will be found to be compounded of the old measurements.
It is impossible to find a point in the body which could not be arrived at by travelling in combinations of the three directions already taken.
But why should space be limited to three independent directions?
Geometers have found that there is no reason why bodies which we can measure should thus be limited. As a matter of fact all the bodies which we can measure are thus limited. So we come to this conclusion, that the space which we use for conceiving ordinary objects in the world is limited to three dimensions. But it might be possible for there to be beings living in a world such that they would conceive a space of four dimensions. 1
It is possible to say a great deal about space of higher dimensions than our own, and to work out analytically many problems which suggest themselves. But can we conceive four-dimensional space in the same way in which we can conceive our own space? Can we think of a body in four dimensions as a unit having properties in the same way as we think of a body having a definite shape in the space with which we are familiar?
There is really no more difficulty in conceiving four-dimensional shapes, when we go about it in the right way, than in conceiving the idea of solid shapes, nor is there any mystery at all about it.
When the faculty to apprehend in four dimensions is acquired—or rather when it is brought into consciousness—for it exists in every one in imperfect form—a new horizon opens. The mind acquires a development of power, and in this use of ampler space as a mode of thought, a path is opened by using that very truth which, when first stated by Kant, seemed to close the mind within such fast limits. Our perception is subject to the condition of being in space. But space is not limited as we at first think.
The next step after having formed this power of conception in ampler space, is to investigate nature and see what phenomena are to be explained by four-dimensional relations.
The thought of past ages has used the conception of a three-dimensional space, and by that means has classified many phenomena and has obtained rules for dealing with matters of great practical utility. The path which opens immediately before us in the future is that of applying the conception of four-dimensional space to the phenomena of nature, and of investigating what can be found out by this new means of apprehension. . . .
For development of knowledge it is necessary to separate the self-elements, i.e., the personal elements which we put in everything cognized by us, from that which is cognized, in order that our attention may not be distracted (upon ourselves) from the properties which we, in substance, perceive.
Only by getting rid of the self-elements in our receptivity do we put ourselves in a position in which we can propound sensible questions. Only by getting rid of the notion of a circular motion of the sun around the earth (i.e., around us—self-element) do we prepare our way to study the sun as it really is.
But the worst about a self-element is that its presence is never dreamed of till it is got rid of.
In order to understand what the self-element in our receptivity means, imagine ourselves to be translated suddenly to another part of the universe, and to find there intelligent beings and to hold conversation with them. If we told them that we came from this world, and were to describe the sun to them, saying that it was a bright, hot body which moved around us, they would reply: "You have told us something about the sun, but you have also told us something about yourselves.". . .
Therefore, desiring to tell something about the sun, we shall first of all get rid of the self-element which is introduced into our knowledge of the sun by the movement of the earth, upon which we are, round it. . . .
One of our serious pieces of work will be to get rid of the self-elements in the knowledge of the arrangement of objects.
The relations of our universe or our space with regard to the wider universe of four-dimensional space are altogether undetermined. The real relationship will require a great deal of study to apprehend, and when apprehended will seem as natural to us as the position of the earth among the other planets seems to us now. . . .
I would divide studies of arrangement into two classes: those which create the faculty of arrangement, and those which use it and exercise it. Mathematics exercises it, but I do not think it creates it; and unfortunately, in mathematics as it is now often taught, the pupil is launched into a vast system of symbols: the whole use and meaning of symbols (namely, as means to acquire a clear grasp of facts) is lost to him. . . .
Of the possible units which will serve for the study of arrangement, I take the cube; and I have found that whenever I took any other unit I got wrong, puzzled, and lost my way. With the cube one does not get along very fast, but everything is perfectly obvious and simple, and builds up into a whole of which every part is evident. . . .
Our work then will be this: a study, by means of cubes, of the facts of arrangement; and the process of learning will be an active one of actually putting up the cubes. Thus we will bring our minds into contact with nature.
(A New Era of Thought.)
Taking all these things into consideration, we should try to define clearly our understanding of those sides of our receptivity dealt with by Kant.
What is space?
Taken as object, that is, perceived by our consciousness, space is for us the form of the universe or the form of the matter in the universe.
Space possesses an infinite extension in all directions. But it can be measured in only three directions independent of one another—in length, breadth, and height; these directions we call the dimensions of space, and we say that our space has three dimensions: it is three-dimensional.
By independent direction we mean in this case a line at right angles to another line.
Our geometry (or the science of measurement of the earth, or matter in space) knows only three such lines, which are mutually at right angles to one another and not parallel among themselves.
But why three only, and not ten or fifteen?
This we do not know.
And here is another very significant fact: either because of some mysterious property of the universe, or because of some mental limitation, we cannot even imagine to ourselves more than three independent directions.
But we speak of the universe as infinite, and because the first condition of infinity is infinity in all directions and in all possible relations, so we must presuppose in space an infinite number of dimensions: that is, we must presuppose an infinite number of lines perpendicular and not parallel to each other; and yet out of these lines we know, for some reason, only three.
It is usually in some such guise that the question of higher dimensionality appears to normal human consciousness.
Since we cannot construct more than three mutually independent perpendiculars, and if the three-dimensionality of our space is conditional upon this, we are forced to admit the indubitable fact of the limitedness of our space in relation to geometrical possibilities: though of course if the properties of space are created by some limitation of consciousness, then the limitedness lies in ourselves.
No matter what this limitedness depends on, it is a fact that it exists.
A given point can be the vertex of only eight independent tetrahedrons. Through a given point it is possible to draw only three perpendicular and not parallel straight lines.
Upon this as a basis, we define the dimensionality of space by the number of lines it is possible to draw in it which are mutually at right angles one with another.
The line upon which there cannot be a perpendicular, that is, another line, constitutes linear, or one-dimensional space.
Upon the surface two perpendiculars are possible. This is superficial, or two-dimensional space.
In "space" three perpendiculars are possible. This is solid, or three-dimensional space.
The idea of the fourth dimension arose from the assumption that in addition to the three dimensions known to our geometry there exists still a fourth, for some reason unknown and inaccessible to us, i.e., that in addition to the three known to us, a mysterious fourth perpendicular is possible.
This assumption is practically founded on the consideration that there are things and phenomena in the world undoubtedly really existing, but quite incommensurable in terms of length, breadth and thickness, and lying as it were outside of three-dimensional space.
By really existing we understand that which produces definite action, which possesses certain functions, which appears to be the cause of something else.
That which does not exist cannot produce any action, has no function, cannot be a cause.
But there are different modes of existence. There is physical existence, recognized by certain sorts of actions and functions, and there is metaphysical existence, recognized by its actions and its functions.
A house exists, and the idea of good and evil exists. But they do not exist in like manner. One and the same method of proof of existence does not suffice for the proof of the existence of a house and for the proof of the existence of an idea. A house is a physical fact, an idea is a metaphysical fact. Physical and metaphysical facts exist, but they exist differently.
In order to prove the idea of a division into good and evil, i.e., a metaphysical fact, I have only to prove its possibility. This is already sufficiently established. But if I should prove that a house, i.e., a physical fact, may exist, it does not at all mean that it exists really. If I prove that a man may own the house it is no proof that he owns it.
Our relation to an idea and to a house are quite different, It is possible by a certain effort to destroy a house—to burn, to wreck it. The house will cease to exist. But suppose you attempt to destroy, by an effort, an idea. The more you try to contest, argue, refute, ridicule, the more the idea is likely to spread, grow, strengthen. And contrariwise, silence, oblivion, non-action, "non-resistance" will exterminate, or in any case will weaken the idea. Silence, oblivion, will not wreck a house, will not hurt a stone. It is clear that the existence of a house and that of an idea are quite different existences.
Of such different existences we know very many. A book exists, and also the contents of a book. Notes exist, and so does the music that the notes combine to make. A coin exists, and so does the purchasing value of a coin. A word exists, and the energy which it contains.
We discern on the one hand, a whole series of physical facts, and on the other hand, a series of metaphysical facts.
As facts of the first kind exist, so also do facts of the second kind exist, but differently.
From the usual positivist point of view it will seem naive in the highest degree to speak of the purchasing value of a coin separately from the coin; of the energy of a word separately from the word; of the contents of a book separately from the book, and so on. We all know that these are only "what people say," that in reality purchasing value, energy of a word, and contents of a book do not exist, that by these conceptions we only denote a series of phenomena in some way linked with coin, word, book, but in substance quite separate from them.
But is it so?
We decided to accept nothing as given, consequently we shall not negate anything as given.
We see in things, in addition to what is external, something internal. We know that this internal element in things constitutes a continuous part of things, usually their principal substance. And quite naturally we ask ourselves, where is this internal element, and what does it represent in and by itself. We see that it is not embraced within our space. We begin to conceive of the idea of a "higher space" possessing more dimensions than ours. Our space then appears to be somehow a part of higher space, i.e., we begin to believe that we know, feel, and measure only part of space, that part which is measurable in terms of length, width and height.
As was said before, we usually regard space as a form of the universe, or as a form of the matter of the universe. To make this clear it is possible to say that a "cube" is the form of the matter in a cube; a "sphere" is the form of the matter in a sphere; "space"—an infinite sphere—is the form of the entire matter of the universe.
H. P. Blavatsky, in The Secret Doctrine, has this to say about space:
The superficial absurdity of assuming that Space itself is measurable in any direction is of little consequence. The familiar phrase (the fourth dimension of space) can only be an abbreviation of the fuller form—the "Fourth dimension of Matter in Space.". . . The progress of evolution may be destined to introduce us to new characteristics of matter. . . ." 1
But the formula defining "space" as "the form of matter in the universe" suffers from this deficiency, that there is introduced in it the concept of "matter," i.e., the unknown.
I have already spoken of that "dead-end siding," x=y, y=x, to which all attempts at the physical definition of matter inevitably lead.
Psychological definitions lead to the same thing.
In a well-known book, The Psychology of the Soul, A. I. Herzen says:
We call matter everything which directly or indirectly offers resistance to motion, directly or indirectly produced by us, manifesting a remarkable analogy with our passive states.
And we call force (motion) that which directly or indirectly communicates movement to us or to other bodies, thus manifesting the greatest similitude to our active states.
Consequently, "matter" and "motion" are something like projections of our active and passive states. It is clear that it is possible to define the passive state only in terms of the active, and the active in terms of the passive—again two unknowns, defining one another.
E. Douglas Fawcett, in an article entitled Idealism and the Problem of Nature in The Quest (April, 1910), discusses matter from this point of view.
Matter (like force) does not give us any trouble. We know all about it, for the very good reason that we invented it. By "matter" we think of sensuous objects. It is mental change of concrete but too complicated facts, which are difficult to deal with.
Strictly speaking, matter exists only as a concept. Truth to tell, the character of matter, even when treated only as a conception, is so unobvious that the majority of persons are unable to tell us exactly what they mean by it.
An important fact is here brought to light: matter and force are just logical concepts, i.e., only words accepted for the designation of a lengthy series of complicated facts. It is difficult for us, educated almost exclusively along physical lines, to understand this clearly, but in substance it may be stated as follows: Who has seen matter and force, and when? We see things, see phenomena. Matter, independently of the substance from which a given thing is made, or of which it consists, we have never seen and never shall see; but the given substance is not quite matter, this is wood, or iron or stone. Similarly, we shall never see force separately from motion. What does this mean? It means that "matter" and "force" are just such abstract conceptions as "value" or "labor," as "the purchasing value of a coin" or the "contents" of a book; it means that matter is "such stuff as dreams are made of." And because we can never touch this "stuff" and can see it only in dreams, so we can never touch physical matter, nor see, nor hear, nor photograph it, separately from the object. We cognize things and phenomena which are bad or good, but we never cognize "matter" and "force" separately from things and phenomena.
Matter is as much an abstract conception as are truth, good and evil.
It is as impossible to put matter or any part of matter into a chemical retort or crucible as it is impossible to sell "Egyptian darkness" in vials. However as it is said that "Egyptian darkness" is sold as a black powder in Athos, or elsewhere, therefore perhaps somewhere, by some one, even matter has been seen.
In order to discuss questions of this order a certain preparation is necessary, or a high degree of intuition; but unfortunately it is customary to consider fundamental questions of cosmogony very lightly.
A man easily admits his incompetency in music, dancing, or higher mathematics, but he always maintains the privilege of having an opinion and being a judge of questions relating to "first principles."
It is difficult to discuss with such men.
For how will you answer a man who looks at you in perplexity, knocks on the table with his fingers and says, "This is matter. I know it; feel! How can it be an abstract conception?" To answer this is as difficult as to answer the man who says: "I see that the sun rises and sets!"
Returning to the consideration of space, we shall under no circumstances introduce unknown quantities in the definition of it. We shall define it only in terms of those two data which we decided to accept at the very beginning.
The world and consciousness are the facts which we decided to recognize as existing.
By the world we mean the combination of all the causes of our sensations in general.
By the material world we mean the combination of causes of a definite series of sensations: those of sight, hearing, touch, smell, taste, sensations of weight, and so on.
Space is either a property of the world or a property of our knowledge of the world.
Three-dimensional space is either a property of the material world or a property of our receptivity of the material world.
Our inquiry is confined to the problem: how shall we approach the study of space?
25:1 Italics by P. D. Ouspensky. Transl.
31:1 "The Secret Doctrine," The Theosophical Publishing Society. Third Edition, p. 271, vol. I.
32:1 This is irony which the English speaking may easily fail to understand. Some unscrupulous monks of the monastery of Athos, famous throughout Greece and Russia, made a practice, it is said, of selling "Egyptian darkness" in little vials, thus making capital out of the credulity and piety of the illiterate Russian pilgrims who were wont to visit this monastery in great numbers. Transl.
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