exhaustive (as will be seen in our own case) criterion of a concept of space and time belonging to the first or the second class, is the answer to the question whether physical space (and time) can or cannot be deformed, i.e., whether or not it is possible to find physical space in which different axioms hold for different places. The answer given by the first class—the Kantian idealists—is in the negative; space and time, they say, are “forms of our intuition,” independent of the external world, and pre-existing to all perception, and they can be only such as are given us; it is not possible for our intuition of the properties of space to be dependent upon our physical theories. The answer of the second class is in the positive: space is a physical fact, the properties of which can be ascertained (measured) by physical means: that is the attitude adopted by the empiricists, sensationalists and the whole of the Mach and Einstein school. Our own reply is that the question has no meaning, because space, according to our definition, can have as few physical or metrical properties as, for instance, the alphabetical order of the vowel-sounds; nevertheless, our point of view does not result in the rejection of the Einstein theory which, in my opinion, can be reconstructed on the basis of our conception of space and time in a manner more satisfactory from the noetic standpoint than the theory is at present. Within the space of the present essay we can only briefly justify our point of view.
43. The fundamental shortcoming of the idealistic point of view,[1] and the cause of its failure lies in its complete irrelevance to the whole question: the decision as to the validity of the laws (postulates) of, e.g., Euclidean or non-Euclidean geometry for physical experience does not depend upon the answer to the question whether this geometry is given us by intuition or by the external fact, but upon the answer to the question what this geometry—irrespective of the way it is given—is: the problem of the structure (metrical properties) of the space in which we place our physical experience remains unaffected whether this space be inside or outside the mind. It is for this reason that Kant’s point of view as to the form of physical space cannot differ from that of Riemann or Einstein.
44. The second standpoint—the classical formulation of which we find in Riemann’s “Habilitazionsschrift”—starts
- ↑ I do not know whether of Kant himself; my criticisms are aimed at those Kantians who hold that geometry is determined by the properties of space, or, better, that geometrical postulates determine or express these properties, and that they, together with the intuition of space, are given us a priori.
