Impossibility of a mathematical definition of dimensions. Why does mathematics not feel dimensions? The entirely conventional character of the designation of dimensions by powers. The possibility of representing all the powers on a line. Kant and Lobachevsky. The difference between non-Euclidean geometry and metageometry. Where should we seek the explanation of the three-dimensionality of the world, if Kant's ideas are correct? Are not the three-dimensional conditions of the world to be found in our perceiving apparatus, in our mind?
Now that we have examined the 'relations which our space itself bears within it' we must return to the question: what really are the dimensions of spaceand why are there three of them?
What must strike us as most strange is the fact that it is impossible to define three-dimensionality mathematically.
We are not clear about this, and it seems a paradox to us, because we always speak of measuring space; nevertheless, it is a fact that mathematics does not feel the dimensions of space.
The question arises, how can such a fine instrument of analysis as mathematics is, not feel dimensions if they constitute certain real properties of space?
In speaking of mathematics, it is necessary, first of all, to accept as a fundamental premise that/or every mathematical expression there is a corresponding relation of certain realities. If this is absent, if this is not so -then there is no mathematics. Expressingthe relations of magnitudes is the task of mathematics; this is its main essence, its chief content. But relations must be between something. It should always be possible to substitute some reality for the algebraical a, b and c. This is the ABC of all mathematics; a, b and c are banknotes: they may be genuine, if they have something real behind them, or they may be counterfeit, if behind them there is no reality.
'Dimensions' play here a very curious role. If we designate them by the algebraic symbols, a, b and c, these symbols will have the character of counterfeit banknotes: they cannot be replaced by any real magnitudes capable of expressing the relations of dimensions.
Usually, dimensions are designated by powers - the first, the second, the third. That is to say, if a line is called a, then the square, the sides of which are equal to this line will be a2, and the cube, the sides of which are equal to this square, will be a3.
As a matter of fact this is what provided Hinton with a basis for his theoryof tessaracts, or four-dimensional solids -a4. But this is sheer fantasy,because, in the first place, the designation of dimensions by powers is purely conventional. All powers may be represented on a line. Let us take a 5millimetre segment of the line a. Then a 25-millimetre segment will be its square, or a2; and a 125-millimetre segment will be its cube, or a3.
How are we to understand that mathematics does not feel dimensions, i.e. that the difference between dimensions cannot be expressed mathematically?
It can be understood and explained in one way only, namely, by the fact that this difference does not exist.
Of course we know that all the three dimensions are actually identical, i.e. that each of the three dimensions in its turn may be regarded as the first, the second, the third, or vice versa. This by itself proves clearly that dimensions are not mathematical magnitudes. All the real properties of a thing can be expressed mathematically as magnitudes, i.e. as numbers showing the relation of these properties to other properties.
In the question of dimensions, however, mathematics seems to see more, or farther, than we do; certain boundaries which stop us do not seem to hinder mathematics from looking through them and seeing that there are no realities to correspond to our concepts of dimensions.
If the three dimensions really corresponded to the three powers, we should have the right to say that only three powers refer to geometry, and that all the other relations between higher powers, beginning from the fourth, lie beyond geometry.
But we have not even got the right to say that. The designation of dimensions by powers is absolutely conventional.
Or, it would be more correct to say that, from the point of view of mathematics, geometry is an artificial construction for the purpose of solving problems based on conditional data, probably deduced from the characteristics of our mentality.
Hinton calls the system of investigation of 'higher space', meta-geometry, and he connects the names of Lobachevsky, Gauss and other investigators of non-Euclidean geometry with metageometry.
Let us now examine how the theories of these scientists stand in relation to the questions we have raised.
Hinton deduces his ideas from Kant and Lobachevsky.
Others, on the contrary, set Kant's ideas in opposition to those of Lobachevsky. Thus, Roberto Bonola, in Non-Euclidean Geometry, says that Lobachevsky's view of space is opposed 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 experimental sciences!*
Which view is correct and in what relation do Lobachevsky's ideas stand to our problem? The most correct answer would be: in no relation. Non-Euclidean geometry is not metageometry, and non-Euclidean geometry stands to metageometry in the same relation as does Euclidean geometry.
The results of all non-Euclidean geometry, which revalued the fundamental axioms of Euclid and found its fullest expression in the works of Bolyai, Gauss and Lobachevsky, are expressed in the formula:
The axioms of a given geometry express the properties of a given space.
Thus, plane geometry accepts all three Euclidean axioms, i.e.:
1 A straight line is the shortest distance between two points.
2 Any figure may be transferred to another place without interfering with its properties.
3 Parallel lines do not meet. (This last axiom is usually formulated differently according to Euclid.)
In the geometry of a sphere or a concave surface only the first two axioms are true, for the meridians, parallel at the equator, meet at the poles.
In the geometry of an irregularly curved surface only the first axiom is true; the second (about the transfer of figures) no longer holds good, for a figuretaken from one place of an irregular surface may change when transferred to another place. And the sum of the angles of a triangle may be more or less than two right angles.
Thus, axioms express the difference in the properties of different kinds of surfaces. A geometric axiom is a law of a given surface.

* Roberto Bonola, Non-euclidean Geometry, a Critical and Historical Study of its Development, Open Court Publishing Co., Chicago, 1912, pp. 92 and 93.

But what is a surface?
Lobachevsky's merit lies in the fact that he found it necessary to revise the fundamental concepts of geometry. But he never went so far as to revalue them from Kant's point of view. Yet at the same time, he never argued against Kant in any sense. For Lobachevsky, as a geometrician, a surface was merely a means for the generalization of certain properties upon which one or another geometric system was built, or the means for generalizing the properties of certain given lines. He probably never thought at all about the reality or the unreality of a surface.
Thus, on the one hand, Bonola is quite wrong in ascribing to Lobachevskyviews opposed to those of Kant, and approaching 'sensualism' and 'the current empiricism'; while on the other hand, there are grounds for thinking that Hinton is quite subjective in ascribing to Lobachevsky and Gauss the inauguration of a new era in philosophy.
Non-Euclidean geometry, including Lobachevsky's geometry, bears no relation to metageometry.
Lobachevsky does not go outside the sphere of three dimensions.
Metageometry regards the sphere of three dimensions as a section of higher space. Among the mathematicians, Riemann came closest of all to this idea, for he understood the relation of time to space.
A point of three-dimensional space is a section of a metageometrical line. The lines metageometry deals with cannot be generalized on any surface. This last may be of the greatest importance for the definition of the difference between geometry (both Euclidean and non-Euclidean) and metageometry.Metageometrical lines cannot be regarded as distances between points in our space; neither can we imagine them as forming any figures in our space.
The examination of the possible properties of lines lying outside our space, their angles, and the relations of these lines and angles to the lines, angles, surfaces and solids of our geometry constitutes the subject of metageometry.
Students of non-Euclidean geometry could not bring themselves to relinquish the surface. There is something really tragic in this. See what surfaces Lobachevsky invented in his investigations of the 11th Euclidean postulate (about parallel lines, or about angles formed by a line intersecting two parallel lines). One of his surfaces resembles the surface of the blades of a ventilator;* another, the inner surface of a funnel. Yet he could not bringhimself to abandon the

* Roberto Bonola, Non-Euclidean Geometry.

surface completely, to cast it away once and for all, and imagine that a line need not necessarily be on a surface, i.e. that a series of lines, parallel or almost parallel, cannot be generalized on any surface, not even in three-dimensional space. This explains why, in creating non-Euclidean geometry, he, and a great many other geometricians, were unable to get out of the three-dimensional world.
Mechanics recognize a line in time, i.e. a line which cannot in any possible way be visualized on a surface, or as the distance between two points in space. This line is taken into account in calculations dealing with machinery. But geometry never had anything to do with this line, but always only with its sections.
Now we may return to the question, 'what is space?' and see whether an answer to this question has been found.
An exact definition and explanation of the three-dimensionality of space as a phenomenon of the world would be an answer.
But there is no such answer. As an objective phenomenon, the three-dimensionality of space remains as mysterious and incomprehensible as before. In relation to threedimensionality it is necessary: either to accept it as a datum and add this datum to the two data we established before; or to admit the incorrectness of this whole objective method of reasoning and return to the other method, indicated at the outset.
Then, starting from the two fundamental data -the world and consciousness — it will be necessary to establish whether three-dimensional space is a property of the world or a property of our perception of the world.
Having started with Kant, who asserts that space is the property of the perception of the world by our consciousness, I purposely turned away from this idea and considered space as a property of the world.
With Hinton, I admitted the surmise that our space bears within itself the conditions which allow us to establish its relations to higher space, and on the basis of this surmise I built a whole series of analogies which made clear to us certain things about the questions of space and time and their mutual relations. But, as has already been said, they did not explain anything concerning the main question of the causes of the threedimensionality of space.
The method of analogies is, on the whole, rather disheartening. It makes one walk in a vicious circle. It helps to clear some things, but does not really give a straight answer to anything. After numerous and prolonged attempts to find one's way in complex problems with the help of analogies, one begins to feel the uselessness of all one's efforts; one feels that, with these analogies, one is merely walking alongside a wall -and then, with a feeling of complete hatred and disgust for analogies, one begins to see the necessity for seeking some direct way which will lead straight to where one needs to go.
This problem of higher dimensions has usually been tackled by means of analogies. Only very recently has science begun to work out that direct method which will be detailed later on.
So, if we wish to follow the direct road, without deviating from it, we must rigidly adhere to Kant's fundamental propositions. But if we formulate Hinton's thought from the point of view of these propositions, we shall get the following result: we bear in ourselves the conditions of our space and therefore must find in ourselves the conditions which will enable us to establish the relation between our space and higher space.
In other words, it is in our mentality, in our perceiving apparatus, that we must find the conditions of the world's three-dimensionality. And it is also there that we must discover the conditions of the possibility of a higherdimensional world.
If we set ourselves this task, we shall find we are on the direct road, and we should be able to get an answer to our question; what is space and its threedimensionality?
How are we to approach the solution of this problem?
Quite clearly, through the study of our consciousness and its properties. We shall be free of all analogies and start on the right and direct road towards the solution of the main problem of the subjective or objective character of space, if we decide to examine the mental forms in which we perceive the world, and see whether there is a correspondence between them and the three-dimensional extension of the world. In other words, we must see whether this idea of the three-dimensional extension of the world with its properties is not the outcome of certain properties of our own mentality.

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