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In 1871, the German mathematician Felix Klein used the concept of a projective metric to classify geometries into elliptic, hyperbolic, and parabolic. This chapter deals with the question whether metrical projective geometry can provide a classification of hypotheses concerning physical space. Such philosophers as Bertrand Russell argued that projective geometry provides us with a priori knowledge in Kant's sense, insofar as projective properties are common to all concepts of spaces. However, Russell did not attribute the same status to metrical properties or metrical projective geometry: the former depend on empirical factors; the latter rests upon a definition of distance that must be stipulated arbitrarily. Therefore, he considered Klein's classification a merely technical result. By contrast, Ernst Cassirer attached great philosophical importance to this result for the clarification of the distinction between the general properties of space and the specific axiomatic structures. Following a line of argument that goes back to Helmholtz, Cassirer used Klein's classification to generalize the Kantian notion of space to a system of hypotheses, including both Euclidean and non-Euclidean geometries. This generalization offered one of the clearest examples of Cassirer's interpretation of the notion of the a priori in terms of a range of hypotheses for the use of mathematics in physics.

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