Euclidean and Non-Euclidean Geometries in the Interpretation of Physical Measurements

  • Francesca Biagioli
Part of the Archimedes book series (ARIM, volume 46)


Klein’s classification of geometries by the use of group theory inaugurated a new phase in the debate on the geometry of space. On the one hand, the conclusion of Riemann’s and Helmholtz’s inquiries into the foundations of geometry appeared to be confirmed: Euclidean geometry does not provide us with the necessary presuppositions for empirical measurement, because both Euclidean and non-Euclidean assumptions can be obtained as special cases of a more general system of hypotheses. On the other hand, Helmholtz had believed that he had shown that the free mobility of rigid bodies implied and was implied by a metric of constant curvature, which includes spherical and elliptic geometries. The group-theoretical approach enabled Sophus Lie to disprove Helmholtz’s argument and provide a mathematically sound solution to the same problem. The most challenging argument against Helmholtz’s empiricism, however, was formulated by Henri Poincaré: observation and experiment cannot contradict geometrical assumptions, because the application of geometrical concepts to empirical objects, including the characterization of solid bodies as “rigid,” already presupposes these kinds of assumptions. The present chapter is devoted to the reception of Poincaré’s argument in neo-Kantianism. In particular, I contrast Poincaré’s conclusion that geometrical axioms are conventions with Cassirer’s view that the interpretation of measurements depends on conceptual rules and ultimately on rational rather than conventional criteria. Cassirer relied on the group-theoretical analysis of space to infer such criteria from the relations of geometrical systems to one another.


Constant Curvature Euclidean Geometry Free Mobility Kantian Philosophy Erlangen Program 
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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Francesca Biagioli
    • 1
  1. 1.ZukunftskollegUniversity of KonstanzKonstanzGermany

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