Chapter VII (part 2)

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. p. 75 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. p. 76 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.