Marco Giovanelli

Drawing an explicit parallel with Felix Klein’s account of the arithmetization of nineteenth-century mathematics, Ernst Cassirer maintains that nineteenth-century physics likewise underwent a process of arithmetization that culminated in the twentieth-century. This paper argues that this constitutes an original insight and a unifying thread running through Cassirer’s philosophy of physics over several decades, one that has yet to receive adequate scholarly attention. In contrast to his fellow Marburg neo-Kantians, Hermann Cohen and Paul Natorp, who regarded the continuum as central to modern scientific knowledge, Cassirer treated number-formation as the prototype of scientific concept-formation in general. The number-sequence combines the structuralist view that each individual number has no intrinsic identity apart from its reciprocal relation to all other numbers with the constructivist view that such a relation is a productive relation, generating ordered sequences of all possible numbers. The paper concludes that, for this reason, Cassirer may be said to have defended a form of structural constructivism, rather than structural realism, as is often claimed.

Stöltzner coined the expression ‘Vienna indeterminism’ to describe a philosophical tradition centered on the Viennese physicist Exner, serving as the ‘historical link’ between Mach and Boltzmann, on the one hand, and von Mises and Frank, on the other. During the early 1930s debate on quantum mechanics, there was a ‘rapprochement’ between Vienna indeterminism and Schlick’s work on causality. However, it was Cassirer’s 1936 monograph _Determinismus und Indeterminismus_ that showed a full ‘convergence’ with major tenets of Vienna indeterminism: the fundamentality of statistical laws, the frequency interpretation of probability, and the statistical interpretation of the uncertainty relations. Yet, Cassirer used these conceptual tools to pursue ‘in parallel’ different philosophical goals. While for the Viennese quantum mechanics represented a fatal blow to the already discredited notion of ‘causality,’ for Cassirer it challenged the classical notion of ‘substantiality,’ the ideas of ‘particles’ as individual substances endowed with properties. The paper concludes that this ‘parallel convergence’ is the most striking and overlooked aspect of _Determinismus und Indeterminismus_, serving as the keystone of its argumentative structure.

Planck’s pioneering contributions to special relativity have received less consideration than one might expect in the historiography and philosophy of physics. Although they are celebrated in isolation, they are mostly not understood as integral to an overarching project. This paper aims (a) to provide a historically accurate overview of Planck’s contributions to the early history of relativity that is reasonably accessible to today’s reader, (b) to demonstrate how these contributions can be presented against the background of Planck’s ‘Helmholtzian’ vision of relativistic general dynamics based on the principle of relativity and principle of least action, and (c) to argue that Planck’s general dynamics serves as an illuminating example of the use of ‘principles’ in physics.

In 1909, Ehrenfest published a note in the Physikalische Zeitschrift showing that a Born rigid cylinder could not be set into rotation without stresses, as elements of the circumference would be contracted but not the radius. Ignatowski and Varićak challenged Ehrenfest’s result in the same journal, arguing that the stresses would emerge if length contraction were a real dynamical effect, as in Lorentz’s theory. However, no stresses are expected to arise, according to Einstein’s theory, where length contraction is only an apparent effect due to an arbitrary choice of clock synchronization. Ehrenfest and Einstein considered this line of reasoning dangerously misleading and took a public stance in the Physikalische Zeitschrift, countering that relativistic length contraction is both apparent and real. It is apparent since it disappears for the comoving observer, but it is also real since it can be experimentally verified. By drawing on his lesser-known private correspondence with Varićak, this paper shows how Einstein used the Ehrenfest paradox as a tool for an ‘Einsteinian pedagogy’. Einstein’s argumentative stance is contrasted with Bell’s use of the Dewan-Beran thread-between-spaceships paradox to advocate for a ‘Lorentzian pedagogy’. The paper concludes that the disagreement between the two ways of ‘teaching special relativity’ stems from divergent interpretations of philosophical categories such as ‘reality’ and ‘appearance’.

The quest for a ‘unified field theory’, which aims to integrate gravitational and electromagnetic fields into a single field structure, spanned most of Einstein’s professional life from 1919 until his death in 1955. It is seldom noted that Hans Reichenbach was possibly the only philosopher who could navigate the technical intricacies of the various unification attempts. By analyzing published writings and private correspondences, this paper aims to provide an overview of the Einstein-Reichenbach relationship from the point of view of their evolving attitudes toward the program of unifying electricity and gravitation. The paper concludes that the Einstein-Reichenbach relationship is more complex than usually portrayed. Reichenbach was not only the indefatigable ‘defender’ of relativity theory but also the caustic ‘attacker’ of Einstein’s and others’ attempts at unified field theory. Over the years, Reichenbach managed to provide the first, and possibly only, overall philosophical reflection on the unified field theory program. Thereby, Reichenbach was responsible for bringing to the debate, often for the first time, some of the central issues of the philosophy of space-time physics: (a) the relation between a theory’s abstract geometrical structures (metric, affine connection) and the behavior of physical probes (rods and clocks, free particles, and so on); (b) the question of whether such association should be regarded as a geometrization of physics or a physicalization of geometry; (c) the interplay between geometrization and unification in the context of a field theory.

In his Zur Einstein’schen Relativitätstheorie, Ernst Cassirer presents relativity theory as the last manifestation of the tradition of the “physics of principles” that, starting from the nineteenth century, has progressively prevailed over that of the “physics of models.” In particular, according to Cassirer, the relativity principle plays a role similar to the energy principle in previous physics. In this article, I argue that this comparison represents the core of Cassirer’s neo-Kantian interpretation of relativity. Cassirer pointed out that before and after Kant, the history of physics presents significant instances in which the search for formal conditions that the laws of nature must satisfy preceded and made possible the direct search for such laws. In his earlier years, Cassirer seems to have regarded principles like the energy principle, the relativity principle, and the principle of least action as a constitutive but provisional form of a priori, imposing specific limitations on the form of the allowable laws of nature. Only in his later years, by attributing an autonomous status to these statements of principle, did Cassirer attribute a definitive but merely regulative meaning to the a priori. This does not impose specific requirements on natural laws but only a motivation to search for them.

At the turn of the twentieth century, Helm and Ostwald were the most prominent supporters of so-called ‘energetics’, which aimed to unify all physics by employing the sole concept of energy, without relying on mechanical models. This paper argues that Cassirer's interest in the history of the energy principle and the energetic controversy is entangled with the main themes of his philosophy of physics up to the 1920s: the opposition between the a priori and the a posteriori and the substance-concept and the function-concept. These interwoven motifs are not always easy to disentangle. The paper suggests that Cassirer's interpretation of the energy principle can serve as a guiding thread that runs through Cassirer's philosophy of physics up to the 1920s.

This study reconstructs the 1928–1929 correspondence between Reichenbach and Einstein about the latter’s latest distant parallelism-unified field theory, which attracted considerable public attention at the end of the 1920s. Reichenbach, who had recently become a Professor in Berlin, had the opportunity to discuss the theory with Einstein and therefore sent him a manuscript with some comments for feedback. The document has been preserved among Einstein’s papers. However, the subsequent correspondence took an unpleasant turn after Reichenbach published a popular article on distant parallelism in a newspaper. Einstein directly wrote to the Editorial Board complaining about Reichenbach’s unfair use of off-the-record information. While Reichenbach’s reply demonstrates a sense of personal betrayal at Einstein’s behavior, his published writings of that period point to a sense of intellectual betrayal of their shared philosophical ideals. In his attempts to unify both electricity and gravitation, Einstein had abandoned the physical heuristic that guided him to the relativity theory, to embrace a more speculative, mathematical heuristic that he and Reichenbach had both previously condemned. A decade-long personal and intellectual friendship grew fainter and then never recovered. This study, relying on archival material, aims to revisit the Reichenbach–Einstein relationship in the late 1920s in light of Reichenbach’s neglected contributions to the epistemology of the unified field theory program. Thereby, it hopes to provide a richer account of Reichenbach’s philosophy of space and time.

In his 1916 review paper on general relativity, Einstein made the often-quoted oracular remark that all physical measurements amount to a determination of coincidences, like the coincidence of a pointer with a mark on a scale. This argument, which was meant to express the requirement of general covariance, immediately gained great resonance. Philosophers such as Schlick found that it expressed the novelty of general relativity, but the mathematician Kretschmann deemed it as trivial and valid in all spacetime theories. With the relevant exception of the physicists of Leiden (Ehrenfest, Lorentz, de Sitter, and Nordström), who were in epistolary contact with Einstein, the motivations behind the point-coincidence remark were not fully understood. Only at the turn of the 1960s did Bergmann (Einstein’s former assistant in Princeton) start to use the term ‘coincidence’ in a way that was much closer to Einstein’s intentions. In the 1980s, Stachel, projecting Bergmann’s analysis onto his historical work on Einstein’s correspondence, was able to show that what he started to call ‘the point-coincidence argument’ was nothing but Einstein’s answer to the infamous ‘hole argument’. The latter has enjoyed enormous popularity in the following decades, reshaping the philosophical debate on spacetime theories. The point-coincidence argument did not receive comparable attention. By reconstructing the history of the argument and its reception, this paper argues that this disparity of treatment is not justified. This paper will also show that the notion that only coincidences are observable in physics marks every critical step of Einstein’s struggle with the meaning of coordinates in physics.

In his 1920 monograph Relativitätstheorie und Erkenntnis apriori the young Reichenbach distinguished between two meanings of the a priori: ‚apodictically valid, true for all time‘ and ‚constituting the concept of object‘. At the end of the 1990s Michael Friedman drew again the attention of philosophers of science to this forgotten distinction. In the spirit of Reichenbach’s early Kantianism Friedman attempted to construct a relativized or temporally variable a priori, which is nevertheless constitutive of the object of knowledge. Friedman rejects an alternative historicized version of the a priori elaborated by the Marburg school and in particular by Cassirer. According to Friedman, Cassirer defended a regulative, but absolute version of the a priori, the existence of a yet-to-be-found set of final principles that are conditions of all scientific experience. This paper suggests that using the constitutive/regulative distinction as a basis for comparison is misleading. In order to understand the Marburg school’s conception of the a priori one should get back to Hermann Cohen’s interpretation of Kant and in particular to his own distinction between two meanings of the a priori. A more suitable comparison is that between Cohen’s opposition metaphysical-vs.-transcendental a priori and Reichenbach’s distinction apodictic-vs.-constitutive a priori. If the comparison is conducted along these lines—as already suggested in the mid-1920 s by the Dutch neo-Kantian Alfred C. Elsbach—it turns out that Cohen and the Marburg school and not Reichenbach provided a good example of a relativized a priori.

Matthias Neuber’s book represents an important contribution to the relatively young discipline of the History of Philosophy of Science. Starting roughly in the 1980s, increasing attention has been devoted not only to the relationship between philosophy and the history of science, but to an accurate historical reconstruction of earlier projects within philosophy of science. One of the most outstanding results of these investigations has probably been the radical reshaping of the rather caricatural image of logical empiricism—for better or worse the core of the philosophical heritage of many philosophers of science—summarized in the so-called ‘standard view’. By analyzing the historical, sociological, and philosophical questions surrounding logical empiricism new light has been shed on the sense of a cultural, social, and political mission that characterized it before its emigration from Europe to North America in the 1930s and 1940s. What came to be known as ..

Gerald Holton has famously described Einstein’s career as a philosophical “pilgrimage”. Starting on “the historic ground” of Machian positivism and phenomenalism, following the completion of general relativity in late 1915, Einstein’s philosophy endured (a) a speculative turn: physical theorizing appears as ultimately a “pure mathematical construction” guided by faith in the simplicity of nature and (b) a realistic turn: science is “nothing more than a refinement ”of the everyday belief in the existence of mind-independent physical reality. Nevertheless, Einstein’s mathematical constructivism that supports his unified field theory program appears to be, at first sight, hardly compatible with the common sense realism with which he countered quantum theory. Thus, literature on Einstein’s philosophy of science has often struggled in finding the thread between ostensibly conflicting philosophical pronouncements. This paper supports the claim that Einstein’s dialog with Émile Meyerson from the mid 1920s till the early 1930s might be a neglected source to solve this riddle. According to Einstein, Meyerson shared (a) his belief in the independent existence of an external world and (b) his conviction that the latter can be grasped only by speculative means. Einstein could present his search for a unified field theory as a metaphysical-realistic program opposed to the positivistic-operationalist spirit of quantum mechanics.

In 1912, Ernst Cassirer contributed to the special issue of the Kant-Studien that honored Hermann Cohen's retirement—his mentor and teacher, and the recognized founding father of the so-called 'Marburg school' of Neo-Kantianism. In the context of an otherwise rather conventional presentation of Cohen's interpretation of Kant, Cassirer made a remark that is initially surprising. It is “anything but accurate,” he wrote, to regard Cohen's philosophy as focused “exclusively on the mathematical theory of nature,” as is usually done. A reconstruction of the genesis of Cohen's thought, Cassirer continued, would already refute this interpretation. Actually, “[t]he...

The present paper attempts to show that a 1915 article by Erich Kretschmann must be credited not only for being the source of Einstein’s point-coincidence remark, but also for having anticipated the main lines of the logical-empiricist interpretation of general relativity. Whereas Kretschmann was inspired by the work of Mach and Poincaré, Einstein inserted Kretschmann’s point-coincidence parlance into the context of Ricci and Levi-Civita’s absolute differential calculus. Kretschmann himself realized this and turned the point-coincidence argument against Einstein in his second and more famous 1918 paper. While Einstein had taken nothing from Kretschmann but the expression “point-coincidences”, the logical empiricists, however, instinctively dragged along with it the entire apparatus of Kretschmann’s conventionalism. Disappointingly, in their interpretation of general relativity, the logical empiricists unwittingly replicated some epistemological remarks Kretschmann had written before General Relativity even existed.

This paper attempts to explain the emergence of the logical empiricist philosophy of space and time as a collision of mathematical traditions. The historical development of the ``Riemannian'' and ``Helmholtzian'' traditions in 19th century mathematics is investigated. Whereas Helmholtz's insistence on rigid bodies in geometry was developed group theoretically by Lie and philosophically by Poincaré, Riemann's Habilitationsvotrag triggered Christoffel's and Lipschitz's work on quadratic differential forms, paving the way to Ricci's absolute differential calculus. The transition from special to general relativity is briefly sketched as a process of escaping from the Helmholtzian tradition and entering the Riemannian one. Early logical empiricist conventionalism, it is argued, emerges as the failed attempt to interpret Einstein's reflections on rods and clocks in general relativity through the conceptual resources of the Helmholtzian tradition. Einstein's epistemology of geometry should, in spite of his rhetorical appeal to Helmholtz and Poincaré, be understood in the wake the Riemannian tradition and of its aftermath in the work of Levi-Civita, Weyl, Eddington, and others

This paper offers a historical overview of Einstein׳s vacillating attitude towards ‘phenomenological’ and ‘dynamical’ treatments of rods and clocks in relativity theory. In Einstein׳s view, a realistic microscopic model of rods and clocks was needed to account for the very existence of measuring devices of identical construction that always measure the same unit of time and the same unit of length. It will be shown that the empirical meaningfulness of both relativity theories depends on what, following Max Born, one might call the ‘principle of the physical identity of the units of measure’. In an attempt to justify the validity of such a principle, Einstein was forced by different interlocutors, in particular Hermann Weyl and Wolfgang Pauli, to deal with the genuine epistemological, rather than the physical question of whether a theory should be required to describe the material devices needed for its own verification.

This paper attempts to show how the logical empiricists’ interpretation of the relation between geometry and reality emerges from a “collision” of mathematical traditions. Considering Riemann’s work as the initiator of a 19th century geometrical tradition, whose main protagonists were Helmholtz and Poincaré, the logical empiricists neglected the fact that Riemann’s revolutionary insight flourished instead in a non-geometrical tradition dominated by the works of Christoffel and Ricci-Curbastro roughly in the same years. I will argue that, in the attempt to interpret general relativity as the last link of the chain Riemann–Helmholtz–Poincaré–Einstein, logical empiricists were led to argue that Einstein’s theory of gravitation mainly raised a problem of mathematical under-determination, i.e. the discovery that there are physical differences that cannot be expressed in the relevant mathematical structure of the theory. However, a historical reconstruction of the alternative Riemann–Christoffel–Ricci–Einstein line of evolution shows on the contrary that the main philosophical issue raised by Einstein’s theory was instead that of mathematical over-determination, i.e. the recognition of the presence of redundant mathematical differences that do not have any correspondence in physical reality

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’