The impossibility of the mathematical definition of dimensions. Why does not mathematics sense dimensions? The entire conditionality of the representation of dimensions by powers. The possibility of representing all powers on a line. Kant and Lobachevsky. The difference between non-Euclidian geometry and metageometry. Where shall we find the explanation of the three-dimensionality of the world, if Kant's ideas are true? Are not the conditions of the three-dimensionality of the world confined to our receptive apparatus, to our psyche?
NOW that we have studied those "relations which our space itself bears within it" we shall return to the questions: But what in reality do the dimensions of space represents—and why are there three of them?
The fact that it is impossible to define three-dimensionality mathematically must appear most strange.
We are little conscious of this, and it seems to us a paradox, because we speak of the dimensions of space, but it remains a fact that mathematics does not sense the dimensions of space.
The question arises, how can such a fine instrument of analysis as mathematics not feel dimensions, if they represent some real properties of space?
Speaking of mathematics, it is necessary to recognize first of all, as a fundamental premise, that correspondent to each mathematical expression is always the relation of some realities.
If there is no such a thing, if it be not true—then there is no mathematics. This is its principal substance, its principal contents. To express the correlations of magnitudes is the problem of mathematics. But these correlations must be between something. Instead of algebraical a, b and c it must be possible to substitute some reality. This is the ABC of all mathematics; a, b and c are credit bills; they can be good ones only if behind them there is a real something, and they can be counterfeited if behind them there is no reality whatever.
"Dimensions" play here a very strange rôle. If we designate them by the algebraic symbols a, b and c, they have the character of counterfeit credit bills. For this a, b and c it is impossible to substitute any real magnitudes which are capable of expressing the correlations of dimensions.
Usually dimensions are represented by powers: the first, the second, the third; that is, if a line is called a, then a square, the sides of which are equal to this line, is called a2, and a cube, the face of which is equal to this square, is called a3.
This among other things gave Hinton the foundation on which he constructed his theory of tesseracts, four-dimensional solids—a4. But this is pure fantasy. First of all, because the representation of "dimensions" by powers is entirely conditional. It is possible to represent all powers on a line. For example, take the segment of a line equal to five millimetres; then a segment equal to twenty-five millimetres will be the square of it, i.e., a2 and a segment of one hundred and twenty-five millimetres will be the cube—a3.
How shall we understand that mathematics does not feel dimensions—that it is impossible to express mathematically the difference between dimensions?
It is possible to understand and explain it by one thing only—namely, that this difference does not exist.
We really know that all three dimensions are in substance identical, that it is possible to regard each of the three dimensions either as following the sequence, the first, the second, the third, or the other way about. This alone proves that dimensions are not mathematical magnitudes. All the real properties of a thing can be expressed mathematically as quantities, i.e., numbers, showing the relation of these properties to other properties.
But in the matter of dimensions it is as if mathematics sees more than we do, or farther than we do, through some boundaries which arrest us but not it—and sees that no realities whatever correspond to our concepts of dimensions.
If the three dimensions really corresponded to three powers, then we should have the right to say that only these three powers refer to geometry, and that all the other higher powers, beginning with the fourth, lie beyond geometry.
But even this is denied us. The representation of dimensions by powers is perfectly arbitrary.
More accurately, geometry, from the standpoint of mathematics, is an artificial system for the solving of problems based on conditional data, deduced, probably, from the properties of our psyche.
The system of investigation of "higher space" Hinton calls metageometry, and with metageometry he connects the names of Lobachevsky, Gauss, and other investigators of non-Euclidian geometry.
We shall now consider in what relation the questions touched upon by us stand to the theories of these scientists. Hinton deduces his ideas from Kant and Lobachevsky.
Others, on the contrary, place Kant's ideas in opposition to those of Lobachevsky. Thus Roberto Bonola, in Non-Euclidian Geometry, declares that Lobachevsky's conception of space is contrary to that of Kant. He says:
The Kantian doctrine considered space as a subjective intuition, a necessary presupposition of every experience. Lobachevsky's doctrine was rather allied to sensualism and the current empiricism, and compelled geometry to take its place again among the experienced sciences. 1
Which of these views is true, and in what relation do Lobachevsky's ideas stand to our problem? The correct answer to this question is: in no relation. Non-Euclidian geometry is not metageometry, and non-Euclidian geometry stands in the same relation to metageometry as Euclidian geometry itself.
The results of non-Euclidian geometry, which have submitted the fundamental axioms of Euclid to a revaluation, and which have found the most complete expression in the works of Bolyai, Gauss, and Lobachevsky, are embraced in the formula:
The axioms of a given geometry express the properties of a given space.
Thus geometry on the plane accepts all three Euclidian axioms, i.e.:
1. A straight line is the shortest distance between two points.
2. Any figure may be transferred into another position without changing its properties.
3. Parallel lines do not meet.
(This last axiom is formulated differently by Euclid.)
In geometry on a sphere, or on a concave surface the first two axioms alone are true, because the meridians which are separated at the equator meet at the poles.
In geometry on the surface of irregular curvatures only the first axiom is true—the second, regarding the transference of figures, is impossible because the figure taken in one part of an irregular surface can change when transferred into another place. Also, the sum of the angles of a triangle can be either more or less than two right angles.
Therefore, axioms express the difference of properties of various kinds of surfaces.
A geometrical axiom is a law of given surface.
But what is a surface?
Lobachevsky's merit consists in that he found it necessary to revise the fundamental concepts of geometry. But he never went so far as to revalue these concepts from Kant's standpoint. At the same time he is in no sense contradictory to Kant. A surface in the mind of Lobachevsky, as a geometrician, was only a means for the generalization of certain properties on which this or that geometrical system was constructed, or the generalization of the properties of certain given lines. About the reality or the unreality of a surface, he probably never thought.
Thus on the one hand, Bonola, who ascribed to Lobachevsky views opposite to Kant, and their nearness to "sensualism" and "current empiricism," is quite wrong, while on the other hand, it is not impossible to conceive that Hinton entirely subjectively ascribes to Gauss and Lobachevsky their inauguration of a new era in philosophy.
Non-Euclidian geometry, including that of Lobachevsky, has no relation to metageometry whatever.
Lobachevsky does not go outside of the three-dimensional sphere.
Metageometry regards the three-dimensional sphere as a section of higher space. Among mathematicians, Riemann, who understood the relation of time to space, was nearest of all to this idea.
The point, of three-dimensional space, is a section of a meta-geometrical line. It is impossible to generalize on any surface whatever the lines considered in metageometry. Perhaps this last is the most important for the definition of the difference between geometries (Euclidian and non-Euclidian and metageometry). It is impossible to regard metageometrical lines as distances between points in our space, and it is impossible to represent them as forming any figures in our space.
The consideration of the possible properties of lines lying out of our space, the relation of these lines and their angles to the lines, angles, surfaces and solids of our geometry, forms the subject of metageometry.
The investigators of non-Euclidian geometry could not bring themselves to reject the consideration of surfaces. There is something almost tragic in this. See what surfaces Beltrami invented in his investigations of non-Euclidian geometry—one of his surfaces resembles the surface of a ventilator, another, the inner surface of a funnel. But he could not decide to reject the surface, to cast it aside once and for all, to imagine that the line can be independent of the surface, i.e., a series of lines which are parallel or nearly parallel cannot be generalized on any surface, or even in three-dimensional space.
And because of this, both he and many other geometers, developing non-Euclidian geometry, could not transcend the three-dimensional world.
Mechanics recognizes the line in time, i.e., such a line as it is impossible by any means to imagine upon the surface, or as the distance between two points of space. This line is taken into consideration in the calculations pertaining to machines. But geometry never touched this line, and dealt always with its sections only.
Now it is possible to return to the question: what is space? and to discover if the answer to this question has been found.
The answer would be the exact definition and explanation of the three-dimensionality of space as a property of the world.
But this is not the answer. The three-dimensionality of space as an objective phenomenon remains just as enigmatical and inconceivable as before. In relation to three-dimensionality it is necessary:
Either to accept it as a thing given, and to add this to the two data which we established in the beginning.
Or to recognize the fallacy of all objective methods of reasoning, and return to another method, outlined in the beginning of the book.
Then, on the basis of the two fundamental data, the world and consciousness, it is necessary to establish whether three-dimensional space is a property of the world, or a property of our knowledge of the world.
Beginning with Kant, who affirms that space is a property of the receptivity of the world by our consciousness, I intentionally deviated far from this idea and regarded space as a property of the world.
Along with Hinton, I postulated that our space itself bears within it the relations which permit us to establish its relations to higher space, and on the foundation of this postulate I built a whole series of analogies which somewhat clarified for us the problems of space and time and their mutual co-relations, but which, as was said, did not explain anything concerning the principal question of the causes of the three-dimensionality of space.
The method of analogies is, generally speaking, a rather tormenting thing. With it, you walk in a vicious circle. It helps you to elucidate certain things, and the relations of certain things, but in substance it never gives a direct answer to anything. After many and long attempts to analyze complex problems by the aid of the method of analogies, you feel the uselessness of all your efforts; you feel that you are walking alongside of a wall. Thereupon you begin to experience simply a hatred and aversion for analogies, and you find it necessary to search in the direct way which leads where you need to go.
The problem of higher dimensions has usually been analyzed by the method of analogies, and only very lately has science begun to elaborate that direct method which will be shown later on.
If we desire to go straight, without deviating, we shall keep strictly up to the fundamental propositions of Kant. But if we formulate Hinton's above-mentioned thought from the point of view of these propositions, it will be as follows: We bear within ourselves the conditions of our space, and therefore within ourselves we shall find the conditions which will permit us to establish correlations between our space and higher space.
In other words, we shall find the conditions of the three-dimensionality of the world in our psyche, in our receptive apparatus—and shall find exactly there the conditions of the possibility of the higher dimensional world.
Propounding the problem in this way, we put ourselves upon the direct path, and we shall receive an answer to our question, what is space and its three-dimensionality?
How may we approach the solution of this problem?
Plainly, by studying our consciousness and its properties.
We shall free ourselves from all analogies, and shall enter upon the correct and direct path toward the solution of the fundamental question about the objectivity or subjectivity of space, if we shall decide to study the psychical forms by which we perceive the world, and to discover if there does not exist a correspondence between them and the three-dimensionality of the world—that is, if the three-dimensional extension of space, with its properties, does not result from properties of the psyche which are known to us.
75:1 Roberto Bonola, "Non-Euclidian Geometry." The Open Court Publishing Co., Chicago, 1912, pp. 92, 93.
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