Marco Giovanelli

Einstein’s “point-coincidence argument'” as a response to the “hole argument” is usually considered as an expression of “Leibniz equivalence,” a restatement of indiscernibility in the sense of Leibniz. Through a historical-critical analysis of Logical Empiricists' interpretation of General Relativity, the paper attempts to show that this labeling is misleading. Logical Empiricists tried explicitly to understand the point-coincidence argument as an indiscernibility argument of the Leibnizian kind, such as those formulated in the 19th century debate about geometry, by authors such as Poincaré, Helmholtz or Hausdorff. However, they clearly failed to give a plausible account of General Relativity. Thus the point-coincidence/hole argument cannot be interpreted as Leibnizian indiscernibility argument, but must be considered as an indiscernibility argument of a new kind. Weyl's analysis of Leibniz's and Einstein's indiscernibility arguments is used to support this claim.

This paper analyzes correspondence between Reichenbach and Einstein from the spring of 1926, concerning what it means to ‘geometrize’ a physical field. The content of a typewritten note that Reichenbach sent to Einstein on that occasion is reconstructed, showing that it was an early version of §49 of the untranslated Appendix to his Philosophie der Raum-Zeit-Lehre, on which Reichenbach was working at the time. This paper claims that the toy-geometrization of the electromagnetic field that Reichenbach presented in his note should not be regarded as merely a virtuoso mathematical exercise, but as an additional argument supporting the core philosophical message of his 1928 monograph. This paper concludes by suggesting that Reichenbach’s infamous ‘relativization of geometry’ was only a stepping stone on the way to his main concern—the question of the ‘geometrization of gravitation’

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

The discovery that Einstein's celebrated argument for general covariance, the 'point-coincidence argument ', was actually a response to the ' hole argument ' has generated an intense philosophical debate in the last thirty years. Even if the philosophical consequences of Einstein's argument turned out to be highly controversial, the protagonists of such a debate seem to agree on considering Einstein's argument as an expression of 'Leibniz equivalence', a modern version of Leibniz's celebrated indiscernibility arguments against Newton's absolute space. The paper attempts to show that the reference to Leibniz, however plausible at first sight, is actually in many respects misleading. In particular it is claimed that the Logical Empiricists offer a significant historical example of an attempt to interpret the point-coincidence argument as an indiscernibility argument in the sense of Leibniz, similar to those used in 19th century by Helmholtz, Hausdorff or Poincaré. However the logical empiricist account of General Relativity clearly failed to grasp the philosophical novelty of Einstein's theory. Thus, if Einstein's point coincidence/ hole argument can be regarded as an indiscernibility argument, it cannot be an indiscernibility argument in the sense of Leibniz. Einstein rather introduced a new form of indiscernibility argument, which might be better described as an expression of 'Einstein-equivalence'. Developing some ideas of Weyl it is argued that, whereas Leibniz's arguments introduced the notion of 'symmetry' in the history of science, Einstein's argument seems to anticipate what we now call 'gauge freedom'. If in the first case indiscernibility arises from a lack of mathematical structure, in the second case it is a consequence of a surplus of mathematical structure. _German_ Die Entdeckung, dass Einsteins berühmtes Punkt-Koinzidenz- Argument zur allgemeinen Kovarianz tatsächlich eine Reaktion auf die Lochbetrachtung war, hat in den vergangenen 30 Jahren zu einer intensiven philosophischen Debatte geführt. Auch wenn die philosophischen Konsequenzen äußerst kontrovers gesehen werden, stimmen die Protogonisten doch darin überein, das Argument als Ausdruck von Leibniz-Äquivalenz, mithin als eine moderne Version von Leibniz berühmten Ununterscheidbarkeitsargumenten gegen Newtons absoluten Raum aufzufassen. Ziel des Aufsatzes ist es zu zeigen, dass der Bezug zu Leibniz, wenn auch auf den ersten Blick plausibel, tatsächlich in vielerlei Hinsicht irreführend ist. Insbesondere wird dahingehend argumentiert, dass die Logischen Empiristen ein signifikantes historisches Beispiel für einen Versuch darstellen, das Punkt-Koinzidenz Argument als ein Ununterscheidbarkeitsargument im Sinne von Leibniz, ähnlich denen im 19. Jahrhundert von Helmholtz, Hausdorff und Poincaré vorgebrachten, zu deuten. Dieser Deutung der Allgemeinen Relativitätstheorie gelingt es aber nicht, das eigentlich philosophisch Neue von Einsteins Theorie plausibel zu interpretieren. Wenn Einsteins Punkt-Koinzidenz/Lochargument als ein Ununterscheidbarkeitsargument angesehen werden soll, kann dies kein Argument à la Leibniz sein. Vielmehr hat Einstein ein neuartiges Ununterscheidbarkeitsargument eingeführt, das vielleicht besser als,Einstein-Äquivalenz' charakterisiert werden sollte. Durch Aufnahme und Weiterentwicklung einiger Ideen von Weyl wird gezeigt, dass Leibniz' Argumente zwar das Konzept der,Symmetrie' in die Wissenschaftsgeschichte eingebracht haben, Einsteins Argument aber etwas antizipiert hat, was heute gewöhnlich,Eichfreiheit' genannt wird. Wird im ersten Fall die Ununterscheidbarkeit durch ein zu wenig an mathematischer Struktur erzeugt, so ist sie im zweiten Fall gerade die Folge eines Überschusses an mathematischer Struktur

By inserting the dialogue between Einstein, Schlick and Reichenbach in a wider network of debates about the epistemology of geometry, the paper shows, that not only Einstein and Logical Empiricists came to disagree about the role, principled or provisional, played by rods and clocks in General Relativity, but they actually, in their life-long interchange, 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. Logical Empiricists, though carefully analyzing the Einstein-Weyl debate, tried on the contrary 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 “Helmhotzian” 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, whose philosophical significance, however, still needs to be fully explored.