Immanuel Kant and the Greater Glory of Geometry

Abstract

One of the objections most often raised against Kant’s treatment of the nature and foundations of geometry is that it cannot accommodate the rise of non-euclidean geometry, much less its success in providing a more accurate description of the physical world. Russell, however, argued that it was really the discovery by Hilbert and others of a complete axiomatization of euclidean geometry more than anything else that revealed the irrelevance of Kantian intuition for its foundations. The scholar G. Martin, on the other hand, has claimed that Kant was really the first and true champion of the axiomatic method, and indeed in such a way that by Kant’s lights non-euclidean geometry was not only an inevitable logical possibility, but also not constructible in intuition. To evaluate all these claims it is imperative to determine the level of axiomatic consciousness of geometry actually obtained by Kant, for as Hilbert (1922) himself has well remarked,

to proceed axiomatically ... is simply to think with consciousness of what one is doing. In earlier times, when they did not use the axiomatic method, men believed in various connections with naive dogmatism. Axiomatics removes the naivete, but still leaves us with all the advantages of belief, (p. 161)