An Essay on the Foundations of Geometry,
Cambridge 1897,
In his early philosophical work Russell concerned himself with German idealism, which was at that time propounded in Britain by J.M. Ellis McTaggart, while he studied as well the mathematical innovations of the 19th century. The latter questioned the Kantian view of space – especially in geometry where a developing plethora of new concepts of space, e.g. metric space or Riemannian manifold, were hardly compatible with the Kantian Transcendental Aesthetic in which there exists a perception [Anschauung] of a space such that every space occurs only as a subspace of this space (Critique of Pure Reason A24/B39):
… when we talk of divers spaces, we mean only parts of one and the same space.
In the essay under consideration Russell proposes an alternative version of Kant’s argument that continues the idea of a space given a priori, but only as a “form of externality” and not a concrete perception of a space. About this space three properties can be deduced a priori (constant curvature, finite dimensionality, and existence of straight lines). At the end of his essay Russell recapitulates:
…
In the second chapter, we endeavoured, by a criticism of some geometrical philosophies, to prepare the ground for a constructive theory of Geometry. We saw that Kant, in applying the argument of the Transcendental Aesthetic to space, had gone too far, since its logical scope extended only to some form of externality in general. We saw that Riemann, Helmholtz and Erdmann, misled by the quantitative bias, overlooked the qualitative substratum required by all judgments of quantity, and thus mistook the direction in which the necessary axioms of Geometry are to found. We also rejected Helmholtz’s view that Geometry depends on Physics, because we found that Physics must assume a knowledge of Geometry before it can become possible. …
Proceeding, in the third chapter, to a constructive theory of Geometry, we saw that projective Geometry, which has no reference to quantity, is necessarily true for any form of externality. Its three axioms—homogeneity, dimension, and the straight line—were all deduced from the conception of a form of externality, and, since some such form is necessary to experience, were all declared a priori. In metrical Geometry, on the contrary, we found an empirical element arising out of the alternatives of Euclidean and non-Euclidean space. …
In the present chapter, we completed our proof of the apriority of the projective and equivalent metrical axioms by showing the necessity, for experience, of some form of externality, given by sensation or intuition, and not merely inferred from other data. Without this, we said, a knowledge of diverse but interrelated things, the corner-stone of all experience, would be impossible. …
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