THE THIRD CANON OF THOUGHT
A KEY TO THE ENIGMAS OF THE WORLD
A new view of Kant's problem. Hinton's books. 'Space-sense' and its evolution. A system for developing the sense of the fourth dimension by means of exercises with different coloured cubes. The geometrical concept of space. Three perpendiculars. Why are there only three? Can everything existing be measured by three perpendiculars? Physical and metaphysical facts. Signs of existence. The reality of ideas. The insufficient evidence of the existence of matter and motion. Matter and motion arc only logical concepts, like 'good' and 'evil'.
I have already said that Kant put forward a problem, but he offered no solution to it nor did he indicate any way to its solution. Neither have any of the known commentators, interpreters, followers or opponents of Kant found this solution or the way to it.
I find the first glimmer of a right understanding of Kant's problem, and the first hints as to a possible way to its solution, in the attempts at a new approach to the study of this problem of space and time, connected with the idea of the 'fourth dimension' and the idea of higher dimensions in general. The books of the English writer, C. H. Hinton, A New Era of Thought and The Fourth Dimension, contain an interesting survey of much that has been done in this direction.*
Hinton says, among other things, that commentaries on Kant's ideas usually deal only with their negative side, that is to say, the fact that we can perceive things through the senses, only in time and space, is regarded as an obstacle, preventing us from seeing what things in themselves are actuallylike, not allowing us to know them as they really are, imposing on them something that does not belong to them, something that shuts them off from us.
But [says Hinton], if we take Kant's statement simply as it is [ - not seeing in spatial perception a hindrance to right perception -and say to ourselves that we apprehend by means of space, then it is equally allowable to consider our space-sense] not as a negative condition hindering us from apprehending the world, but as a positive means by which the mind grasps its experience [i.e. by means of which we apprehend the world].
* Hinton has two separate books The Fourth Dimension and A New Era of Thought; there are also three books of popular articles and fiction, Scientific Romances, where he expounds the same ideas.
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. . . . [We must recognize] 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 very 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 [continues Hinton], 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 thoroughgoing education of the space sense. ... It is left to be organized by accident. . . . [And yet a special development of space-sense makes perfectly clear and simple] a whole series of new conceptions. . . .
Fichte, Schelling, Hegel have developed certain tendencies of Kant and have written remarkable books. But the true successors of Kant are Gauss and Lobatchewski.
For if our intuition of space is the means by which we apprehend, then it follows that there may be different kinds of intuitions of space. . . . This intuition of space must be coloured, so to speak, by the conditions (of the mental 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.*
Hinton devised a complicated system for educating and developing space-sense by means of exercises with a series of different coloured cubes. The books already mentioned are devoted to the exposition of this system. In my opinion Hinton's exercises are interesting from the point of view of theory, but can have a practical significance only in those cases where people have the same mental make-up as Hinton.
According to Hinton, his system of mental exercises should, first of all, lead to the development of the ability to visualize things, not as the eye sees them, i.e. not in perspective, but as they are geometrically; for example, they should teach one to visualize the cube from all sides at once. If one acquires this ability of visualization, not in perspective, it should, in its turn, greatly widen the bounds of the activity of our consciousness, thereby creating new concepts and intensifying our capacity for drawing analogies.
* C. H. Hinton, A New Era of Thought, London, George Alien & Unwin, 1910.
Kant established the fact that an expansion of knowledge under the existing conditions of perception will not bring us any nearer to things in themselves. But there are theories asserting that, if desired, it is possible to change the very conditions of perception and in this way approach to the real essence of things. In the above-mentioned books Hinton attempts to unite together the scientific grounds 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 should be thus 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. . . .
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 [of apprehending in four dimensions] is acquired -or rather when it is brought into consciousness, for it exists in everyone 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 conditions 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 the 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.
To expand our apprehension it is important to separate as far as possible the selfelements, i.e. the personal elements introduced by us into everything we apprehend, from that which is being apprehended, so that our attention may not be distracted (onto ourselves) from the properties of what we actually perceive.
Only 'by getting rid of the self-elements' in our perception do 'we put ourselves in a position in which we can propound sensible questions'. Only 'by getting rid of the notion of its circular motion round the earth [i.e. round us -a self-element] do we prepare our way to study the sun.'
The worst about a self-element [in perception] is, that its presence is never dreamed of till it is got rid of. ...
[In order to understand what the self-element in our perception means, let us] 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 a world, and were to describe the sun to them, saying that it was a bright, hot body which moved round us, they would reply: You have told us something about the sun, but you have also told us something about yourselves.
Therefore, if we wish to know something about the sun we must first of all get rid of the self-element introduced into our apprehension of the sun by the motion round it of the earth, on which we are.
'One of our serious pieces of work' in the education and development of space-sense 'will be to get rid of the self-elements in the knowledge of arrangement [of objects]'.
What the relation of our universe, or our space, to the four-dimensional space may be, is 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 does 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 at once launched into a vast system of symbols [without being given the possibility of grasping their meaning and significance].
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 pan is evident. . . .
Our work will then 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. In this way ... we bring . . . [the mind] into contact with nature.*
Now, taking into consideration all that has been said, let us try to establish exactly how we understand those aspects of our perception of which Kant speaks.
What is space?
Taken as an object, i.e. visualized as outside our consciousness, space is for us the form of the universe or the form of matter in the universe.
Space possesses infinite extension in all directions. But, at the same time, we can measure it in three independent directions only: length, breadth and height. We call these directions dimensions of space and say that our space possesses three dimensions, that it is three-dimensional.
By an independent direction we mean, in this case, a line lying at right angles to another line.
Our geometry (i.e. the science of measuring the earth, or matter in space)knows only three such lines which lie simultaneously at right angles to one another and are not parallel in relation to each other.
Why are there only three and not ten or fifteen?
This we do not know.
Moreover, one other fact is significant -either by virtue of some mysterious quality of the universe, or because of the limitations of our mental apparatus, we cannot visualize more than three perpendiculars.
But we say that space is infinite. Therefore, since the first condition of infinity is infinity in all directions and in all possible respects, we must assume that space has an infinite number of dimensions, that is, assume the possibility of an infinite number of lines perpendicular and not parallel to one another. And in addition we have to assume that for some reason we know only three of these lines.
This is the aspect in which the question of higher dimensions presents itself to our ordinary consciousness.
All the same, since we are incapable of constructing more than three perpendiculars, we are forced to admit that, even if the three-dimensionalityof our space is merely conditional, the limitedness of our space as regards geometrical possibilities is an unquestionable fact. But of course, if these properties of space are created by certain attributes of our own, then it follows that the limitation is also in ourselves.
* C. H. Hinton, A New Era of Thought, London, George Alien & Unwin, 1910.
No matter what this limitation depends on, the fact is that it exists.
A given point can be the vertex of only eight independent tetrahedrons. From a given point only three perpendicular and non-parallel lines can be traced.
Starting from this, we determine the dimensionality of space by the number of lines it is possible to trace in it which would lie at right angles to one another.
On a line there cannot be a perpendicular, that is, another line. It is onedimensional space.
On a surface, two perpendiculars are possible. It is two-dimensional space.
In 'space', there are three perpendiculars. It is 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 a fourth, for some reason inaccessible and unknown to us, i.e. that in addition to the three perpendiculars known to us a mysterious fourth perpendicular is possible. In practice this assumption is based on the consideration that the world contains many things and phenomena about whose real existence there can be no doubt, but which are utterly beyond being measured in length, breadth and height and lie, as it were, outside three-dimensional space.
We may take as really existing that which produces a certain action, has certain functions, represents 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 kinds of existence. There is the physical existence, recognized by actions and functions of a certain kind; and there is the 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 inthe same way. One and the same method of proving existence cannot serve to prove the existence of a house and the existence of an idea. A house is a physical fact, an idea is a metaphysical fact. Both the physical and the metaphysical facts exist, but they exist differently.
In order to prove the idea of the division of good and evil - i.e. a metaphysical fact - I must prove its possibility. This will be sufficient. But if I prove that a house, i.e. a physical fact, can exist, it does not at all mean that it actually does exist. To prove that a man can own a house is no proof that he actually owns it.
Moreover, our relation to an idea and to a house is quite different. By means of a certain effort a house can be destroyed - it can be burned or demolished. The house will cease to exist. But try to destroy an idea byeffort. The more you fight against it, the more you argue, refute, ridicule it, the more the idea will grow, spread and gain strength. On the other hand, silence, oblivion, non-doing, 'non-resistance' will annihilate, or at any rateweaken the idea. But silence, oblivion, will not harm a house or a stone. It is clear that the existence of a house and the existence of an idea are different existences.
We know a great many of such different existences. A book exists and the contents of a book exist. Notes exist, and the music they contain exists. A coin exists and the purchasing value of a coin exists. A word exists and the energy contained in it exists.
On the one hand we see a series of physical facts, on the other, a series of metaphysical facts.
There are facts of the first kind and facts of the second kind; they both exist, but they exist differently.
From the ordinary positivist view it will appear very naive 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 this is only 'a manner of speech', that actually the purchasing value, the energy of a word, the contents of a book, have no existence; they are only concepts by means of which we designate a series of phenomena in some way connected with the coin, the word, the book, but really quite separate from them.
But is it so?
We decided not to accept anything as data and therefore we must not reject anything as data.
We see in things not only an outer aspect but an inner content. We know that this inner content constitutes an inalienable part of things, usually their main essence. And quite naturally we ask ourselves where it is and what it represents. We see that this inner content is not in our space. So we conceive the idea of a 'higher space', possessing more dimensions than ours. Our spacethen becomes a pan of a higher space, as it were, i.e. we begin to suppose that we know, sense and measure only a part of space, that part which ismeasurable in length, breadth and height.
It was said earlier that, as a rule, we regard space as the form of the universe or the form of matter in the universe. To make this more clear -it can be said that a 'cube' is the form of matter in a cube; a 'sphere' is the form of matter in a sphere; 'space' -an infinite sphere - is the form of all the matter contained in the universe. In The Secret Doctrine, H. P. Blavatsky says this 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.*
But the formula denning 'space' as the 'form of matter in the universe' suffers from one defect, namely, it introduces the concept of 'matter', i.e. an unknown. I have already spoken of the blind alley, x = y, y = x, to which all attempts at a physical definition of matter lead. Psychological definitions lead to the same. In his well-known book. The Physiology of the Soul, A. I. Hertzen says:
We call matter everything that, directly or indirectly, offers resistance to motion directly or indirectly produced by us, manifesting in this 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, manifesting in this the greatest resemblance to our active states.
Consequently, 'matter' and 'motion' are, as it were, projections of our active and passive stages. It is clear that the passive state can only be defined by means of the active, and the active by means of the passive. The result is once more two unknowns defining one another.
E. Douglas Fawcett puts it very well when he speaks of matter in his article 'Idealism and the Problem of Nature' in The Quest (April 1910):
Matter (like 'Force') does not present any difficulty at all. We know all about it, for the very good reason that we have invented it. . . . 'Matter' is a creation of our conceiving; a mere way of thinking about sensible objects;
a mental substitute for concrete but unmanageably complex facts. . . .
Strictly speaking. Matter exists only as a concept. . . . Truth to tell, the character of Matter, even when treated only as a conception, is so un-obvious, that the majority of persons are unable to tell exactly what they mean by it.
' H. P. Blavatsky, The Secret Doctrine, London and New York, Theosophical Publishing Society, 3rd edn, 1893, vol. 1, p. 271.
One important point is brought out here: matter and force are only logical concepts, i.e. only terms adopted to designate a long series of diverse facts. It is difficult for us, brought up on 'physics', to understand this. But in reality who has ever seen matter or force? We see things, we see phenomena. Matter separately from the substance of which a given thing is made or consists we have never seen and never shall see. And, a given substance is not matter, it is wood, or iron, or stone. In the same way, we shall never see force separately from action. What does this mean? It means that matter and force are concepts just as abstract as 'value' or 'labour', as the 'purchasingvalue' of a coin, as the 'contents' of a book. It means that matter is 'such stuff as dreams are made on'. And, just as we can never touch this 'stuff', and see it only in dreams, so we can never touch, see, hear or photograph physical matter separately from things. Perfectly or imperfectly, we know things and phenomena, but we shall never know matter and force apart from things and phenomena.
Matter is as much an abstract concept as truth, good or evil.
Matter, or any part of matter, cannot be put into a chemical retort or a crucible, just as 'Egyptian Darkness' cannot be sold in small bottles. But they say that 'Egyptian Darkness' in the form of black powder is sold on Mount Athos or elsewhere, so perhaps someone has also seen matter after all.
In order to find the right approach to these questions it is necessary to have a certain preparation or a great inner flair. Unfortunately people embark with too great an ease on discussions about fundamental questions of the structure of the world.
A man readily admits his incompetence in music or in higher mathematics, or in the art of ballet dancing, but he always reserves the right to have an opinion and voice a judgment on questions referring to 'fundamental principles'.
To talk with such people is very difficult.
For, how will you answer a man who looks at you in perplexity, taps his finger on the table and says, 'This is matter, I know, I feel it. How can this be an abstract concept?' It is just as difficult to answer him as it is difficult to answer the man who says: 'But I see for myself that the sun rises and sets!'
To return to the question of space, we must at all events not introduce unknown quantities into its definition. We must define it with the help of the two data we already decided to accept at the very beginning. The world and our inner life are the two facts we decided to recognize as existing.
By the world we mean the combination of the causes of all our sensations in general.
By the material world we mean the combination of the causes of a definite series of sensations, those of sight, hearing, touch, smell, taste, sensations of weight, of mass, and so on.
Space is either a property of the world or a property of our cognition of the world.
Three-dimensional space is either a property of the material world or a property of our perception of the material world.
So the question is this: how must we approach the study of space?