Chapter VII (part 3)

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. p. 77 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. p. 78 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 .