Marco Giovanelli

The article attempts to reconsider the relationship between Leibniz’s and Kant’s philosophy of geometry on the one hand and the nineteenth century debate on the foundation of geometry on the other. The author argues that the examples used by Leibniz and Kant to explain the peculiarity of the geometrical way of thinking are actually special cases of what the Jewish-German mathematician Felix Hausdorff called “transformation principle”, the very same principle that thinkers such as Helmholtz or Poincaré applied in a more general form in their celebrated philosophical writings about geometry. The first two parts of the article try to show that Leibniz’s and Kant’s philosophies of geometry, despite their differences, appear to be preoccupied with the common problem of the impossibility to grasp conceptually the intuitive difference between two figures (such as a figure and its scaled, displaced or mirrored copy). In the third part, it is argued that from the perspective of Hausdorff’s philosophical-geometrical reflections, this very same problem seems to find a more radical application in Helmholtz’s or Poincaré’s thought experiments on the impossibility of distinguishing distorted copies of our universe from the original one. I draw the conclusion that in Hausdorff’s philosophical work, which has received scholarly attention only recently, one can find not only an original attempt to frame these classical arguments from a set-theoretical point of view, but also the possibility of considering the history of philosophy of geometry from an uncommon perspective, where especially the significance of Kant’s infamous appeal to “intuition” can be judged by more appropriate standards

The paper analyses the significance of the modern concept of „symmetry“ for the understanding of the concept of „intuition“ in Kant's philosophy of geometry. A symmetry transformation or automorphism is a structure preserving mapping of the space into itself that leaves all relevant structure intact so that the result is always like the original, in all relevant respects. Hermann Weyl was the first to show that this idea can be drawn on Leibniz's definition of similarity: two figures are similar if they are only distinguishable through the simultaneous perception of them ( comperceptio ): „an automorphism carries a figure into one that in Leibniz's words is ‚indiscernible from it if each of the two figures is considered by itself‘“. The author argues that, under the light of this definition, Leibniz's notion of „comperceptio“ turns out to play a similar role for the notion of „intuition“ in Kant's philosophy of space and time. Both cases are about the „difference between conceptual definition and intuitive exhibition“, as Weyl puts it. This result has on the one hand an exegetical significance: it frees Kant's notion of „intuition“ from every vague reference to the „visualisation“ of geometrical figures; on the other hand a theoretical one: it makes easy to compare Kant's philosophy of space and time with modern developments of sciences, in which as Weyl first showed, the concept of symmetry plays a fundamental role

By inserting the dialogue between Einstein, Schlick and Reichenbach into a wider network of debates about the epistemology of geometry, this paper shows that not only did Einstein and Logical Empiricists come to disagree about the role, principled or provisional, played by rods and clocks in General Relativity, but also that in their lifelong interchange, they never clearly identified the problem they were discussing. Einstein’s reflections on geometry can be understood only in the context of his ”measuring rod objection” against Weyl. On the contrary, Logical Empiricists, though carefully analyzing the Einstein–Weyl debate, tried to interpret Einstein’s epistemology of geometry as a continuation of the Helmholtz–Poincaré debate by other means. The origin of the misunderstanding, it is argued, should be found in the failed appreciation of the difference between a “Helmholtzian” and a “Riemannian” tradition. The epistemological problems raised by General Relativity are extraneous to the first tradition and can only be understood in the context of the latter, the philosophical significance of which, however, still needs to be fully explored