Criticism of Mathematics as Based on Naive Intuition

Abstract

One is in the habit of representing Kant’s theory of space as though its foundation were all of a sudden shattered by the discovery of non-Euclidean geometry. That was not the case. On the one hand, one may adhere to Kant’s theory of space in spite of the discovery of non-Euclidean geometry; formally, this standpoint is irrefutable — just as irrefutable as that of the man who refused to believe in the discovery of America because he had never been there himself, but also just as arbitrary. On the other hand, there were already two facts known in Kant’s time, which by thorough analysis result in just as many arguments against Kant’s theory of space as can be found in non-Euclidean geometry. These facts are: (1) The possibility of a space of more than three dimensions discovered by... Kant.1 The possibility of a space of more than three dimensions proves just as much, of course, as the possibility of a space in which the customary postulate of parallels is not valid.2 (2) The so-called paradox of the symmetrical solids. It is known that Kant successively advanced the possibility of symmetrical, non-congruent, solid bodies as an argument for various incompatible conceptions about the nature of space. We find the following argument presented by Gauss against the theory of space of Kant’s Kritik der reinen Vernunft. “This difference between right and left is completely determined in itself as soon as one has once fixed (arbitrarily) forward and backward in the plane and above and under in relation to both sides of the plane, even though we can communicate our apprehension of this difference to others only ostensively by reference to material objects actually present. Kant had already made both observations. But one cannot understand how this acute philosopher could have believed the first to be a proof of his notion that space is only the form of our outer intuition, since the second proves so clearly the opposite and shows that space must have a real significance independent of our mode of intuition.” 3