Kant and the impossibility of non-euclidean space

Abstract

In this paper, I discuss the problem raised by the non‐Euclidean geometries for the Kantian claim that the axioms of Euclidean geometry are synthetic a priori, and hence necessarily true. I argue that, although the Kantian view of geometry faces a serious challenge from non‐Euclidean geometries, there are some aspects of Kant’s view about geometry that can still be plausible. However, I will not reduce synthetic a priori Euclidean geometry to simply what Strawson1 calls “phenomenal geometry,” where “phenomenal,” in this context for Strawson, loosely means “visualizable” or “picturable.” Phenomenal geometry, in this sense, is the geometry of imagined scenes. If Euclidean geometry is only necessarily true for imagined/visualized objects and their imagined spatial relations, but not for empirical/sensible world, then I take it that Kant was completely wrong about the nature of geometry. In what follows, I try to save some of Kant’s views, without separating geometry from the empirical world and confining it solely into mental imagery.

Necessity of the fundamental laws that govern appearances (phenomena, in the Kantian sense) is central to Kant’s understanding of the notion of “natural science.” According to Kant, these fundamental laws can be necessary only if they are cognized a priori. He writes, in Metaphysical Foundations of Natural Science: