In 1870, Helmholtz presented a thought experiment concerning the hypothetical measurements in a convex mirror and how the displacements in the mirror would appear to us to follow non-Euclidean (hyperbolic) geometry. After the emergence of the group-theoretical approach to geometry in the works of Felix Klein, Sophus Lie, and Henri Poincaré, Helmholtz’s thought experiment has been typically construed in group-theoretical terms by identifying the displacements under consideration as collineations and the properties of space as the invariants of a larger transformation group than the Euclidean group. However, there is less agreement about what properties would count as the most general in such an account of space and weather or not it is possible to give a mathematical description of such a concept at all. Following Helmholtz’s earlier theory of spatial perception, Moritz Schlick suggested that Helmholtz’s account foreshadowed the distinction of the indescribable qualities of spatial perception from the abstract structures, which are the objects of axiomatic geometry. This talk offers a partial defense of the group-theoretical reading of Helmholtz along the lines of Ernst Cassirer in the fourth volume of The Problem of Knowledge of 1940. In order to avoid the problem raised by Schlick, Cassirer relied on the neo-Kantian view of space not so much as an object of geometry, but as a precondition for the possibility of measurement. His reading of Helmholtz suggests that, although the concept of group does not provide a description of space, the modern way to articulate the concept of space in terms of transformation groups was appropriate, insofar as it provided a foundation of metrical geometry.
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