Philosophers and Einstein's Relativity: The Early Philosophical Reception of the Relativistic Revolution (Boston Studies in the Philosophy and History of Science, 342) 303136497X, 9783031364976

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Boston Studies in the Philosophy and History of Science 342

Chiara Russo Krauss Luigi Laino   Editors

Philosophers and Einstein’s Relativity The Early Philosophical Reception of the Relativistic Revolution

Boston Studies in the Philosophy and History of Science Founding Editor Robert S. Cohen

Volume 342

Series Editors Jürgen Renn, Max Planck Institute for the History of Science, Berlin, Germany Lydia Patton, Virginia Tech, Ona, WV, USA Editorial Board Members Thomas F. Glick, Department of History, Boston University, Boston, USA John Heilbron, University of California, Berkeley, UK Diana Kormos-Buchwald, Dept. of Humanities and Social Sciences, California Institute of Technology, Pasadena, CA, USA Agustí Nieto-Galan, Dept. de Filosofia, Edifici B, Universitat Autonoma de Barcelona, Bellaterra (Cerdanyola del V.), Spain Nuccio Ordine, Campus di Arcavacata - Dip. di Filologia, Universitá della Calabria, RENDE, Cosenza, Italy Ana Simões, Department of History and Philosophy of Science, FCUL, Universidade de Lisboa, Lisboa, Portugal John J. Stachel, Boston University, Brookline, MA, USA Baichun Zhang, Inst for the History of Natural Sciences, Chinese Academy of Science, Beijing, China Kostas Gavroglu, University of Athens, Athens, Greece Associate Editor Peter McLaughlin, Department of Philosophy, Universität Heidelberg, Heidelberg, Baden-Württemberg, Germany Managing Editor Lindy Divarci, c/o Divarci, Max Planck Institute for the History of Science, Berlin, Berlin, Germany

The series Boston Studies in the Philosophy and History of Science was conceived in the broadest framework of interdisciplinary and international concerns. Natural scientists, mathematicians, social scientists and philosophers have contributed to the series, as have historians and sociologists of science, linguists, psychologists, physicians, and literary critics. The series has been able to include works by authors from many other countries around the world. The editors believe that the history and philosophy of science should itself be scientific, self-consciously critical, humane as well as rational, sceptical and undogmatic while also receptive to discussion of first principles. One of the aims of Boston Studies, therefore, is to develop collaboration among scientists, historians and philosophers. Boston Studies in the Philosophy and History of Science looks into and reflects on interactions between epistemological and historical dimensions in an effort to understand the scientific enterprise from every viewpoint.

Chiara Russo Krauss • Luigi Laino Editors

Philosophers and Einstein’s Relativity The Early Philosophical Reception of the Relativistic Revolution

Editors Chiara Russo Krauss University of Naples Federico II Naples, Italy

Luigi Laino University of Naples Federico II Naples, Italy

ISSN 0068-0346 ISSN 2214-7942 (electronic) Boston Studies in the Philosophyand History of Science ISBN 978-3-031-36497-6 ISBN 978-3-031-36498-3 (eBook) https://doi.org/10.1007/978-3-031-36498-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

The publication of Einstein’s theory of relativity was an event bound to revolutionize not only the world of physics but also the world of philosophy. During the nineteenth century, all the philosophical schools had developed different strategies to deal with and account for the rise of modern, specialized sciences. Now, these strategies were being put to the test by the appearance of Einsteinian theories, having to prove their capacity to encompass and explain the newest advances in physics. Positivists, neoKantians, phenomenologists, critical realists . . . if they wanted to remain relevant, they all had to elaborate their own interpretation of the theory of relativity. For this reason, the early philosophical discussion about the theory of relativity is important not only for the intrinsic value or validity of the epistemological interpretations of Einstein’s theories developed during that time. Most importantly, the way how philosophers and philosophical schools responded to the new factor that was shaking the foundations of their systems allows us to get further insight into those philosophical systems and—more generally—into the complex relationship between philosophical discourse and scientific advances. This volume contains some of the papers presented during the conference “Philosophers and Einstein’s Relativity”, held online in 2021 from May 26–28, and organized within the project “Scientific Philosophy”, founded by the Italian Ministry of University and Research through the SIR program (Scientific Independence of Young Researchers), and directed by Chiara Russo Krauss. In organizing the conference, we invited scholars from different countries to discuss, first and foremost, the early philosophical reception of relativity theory in the Germanspeaking area. Talks favoured an insightful exchange of ideas that we hope this book can enliven. Moreover, by presenting these works together in one book, the reader can have an insight into the different interpretations of the theory of relativity by philosophers of the first half of the twentieth century, comparing their different strategies for dealing with the theory of relativity, and also understanding how they were responding not only to Einstein’s relativity but also to each other. We decided to begin with the chapter by Klaus Hentschel since it provides a general overview of the epistemological criteria concerning the evaluation of philosophical interpretations of relativity theory. Afterward, we outlined an ideal v

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path from Petzoldt to the Einstein-Reichenbach dispute upon the unified field theory program. At the end of the book, the reader may find those chapters which unfold a natural link to the more contemporary debate. As Hentschel has shown in his pivotal book published in 1990 (Interpretationen und Fehlinterpretationen der speziellen und der allgemeinen Relativitätstheorie), most of the early interpretations of both special and general relativity were misinterpretations. However, some reasons allow such misinterpretations to emerge, and Hentschel will focus on them painstakingly. The results of his analysis amount to a structural matrix (“interpretational frame”) working as the basis of all the second-order (mis)interpretations. In this respect, Hentschel insists that Fleck’s claim that “thought collectives” impact human thinking and interpretation is a reasonable explanation for the propagation of errors. Russo Krauss, with her chapter “A Machian Interpretation of the Theory of Relativity? Joseph Petzoldt’s Reading of Einstein”, focuses on the relation between Joseph Petzoldt and Einstein. Einstein’s sympathy for Mach and, in general, for the phenomenalistic interpretation of relativity has been undermined over the years, also in light of Einstein’s late comments on conventionalism and Kantianism in his autobiographical notes. However, Russo Krauss displays that Petzoldt was the one who upheld that Machian Philosophy naturally fits relativity. Russo Krauss emphasizes Petzoldt’s concept of “Eindeutigkeit”, which constitutes, as Ryckman has noticed, a sort of shared platform for all those who will work on the epistemology of relativity (especially neo-Kantians and neo-empiricists). According to Russo Krauss, Einstein’s endorsement of Petzoldt’s “Eindeutigkeit”—as a deterministic turn departing from Mach’s Humean conception of causality—enables us to understand in which sense one may consider Einstein a phenomenalist. Pecere’s chapter—“The End of Matter? On the Early Reception of Relativity”— addresses another theme arising from the very early interpretation of relativity, that of the end of matter as introduced by Poincaré’s La fin de la matière (1906). Pecere will follow the development of Poincaré’s claim in Marburg neo-Kantianism. For many reasons, Poincaré was one of the most relevant sources for the neo-Kantians, and Pecere aims to show that they exaggerated Poincaré’s take, originally conceived of as a “non-conclusive” hypothesis. The neo-Kantians forced Poincaré’s prudence in virtue of their paradigmatic presentation of the history of idealism, pictured as a more or less linear tradition which goes from Plato to Kant, through Descartes and Leibniz. Pecere concludes that if this attempt does not cram relativity into the neoKantian tradition, it does not even cope with some non-idealistic takes in relativity, such as those regarding the concept of field and energy-matter. Biagioli’s chapter—“Cassirer and Klein on the Geometrical Foundations of Relativistic Physics”—deals with a re-evaluation of the classic judgment upon the limitation of invariance-based approaches concerning relativity theory. It seems that invariance-based frames capture invariants, which in the case of Einstein’s theory are hard to near for its use of Riemannian methods; for this very reason, Biagioli will draw attention to Klein’s subtractive and additive strategies, which are more in line with Riemann’s work. Furthermore, Biagioli suggests that Cassirer’s reliance on Klein in Substance and Function (1910) and his later appreciation of Riemannian geometry concerning general relativity in Einstein’s Theory of Relativity (1921)

Preface

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speak in favour of a natural adaptation of his a priori approach to relativity. By naturally expanding his standpoint, Biagioli exploits Cassirer’s commitment to a variety of geometrical traditions and be able to avoid some of the crudest objections against the relativization of the a priori. Laino, in his chapter “Natorp, Cassirer and the Influence of Relativity on neoKantian Philosophy,” focuses on the often underrated relationship between Natorp and Cassirer concerning relativity theory. Usually, neo-Kantian (Ferrari) and nonneo-Kantian scholars (Hentschel) distinguish Natorp’s reading from Cassirer’s in terms of “immunization” and “revision”, in order to contrast Natorp’s conservative approach with Cassirer’s liberalization of the a priori. However, by exploiting Hentschel’s evaluative and comparative criteria, Laino shows that Natorp’s interpretation is partially justified if compared to the status of physical research in 1910. In particular, Natorp already dealt with the concept of invariance, conceived of as a regulative principle that dominates the history of science and its development. By doing so, Laino does not intend to overlook the difference between Natorp and Cassirer. Indeed, he argues that Natorp is wrong when he reversed the inclusive relationship between classic and relativistic group transformations. Quite the opposite, Cassirer’s interpretation is not affected by such epistemological lack, and, more in general, his philosophy is largely inspired by general covariance as a heuristic principle determining the concept of objectivity. In his chapter “Attacking Einstein. The Reichenbach-Einstein Debate over the Unified Field Theory Program”, Giovanelli reconstructs the debate between Einstein and Reichenbach about the unification of electricity and gravitation. He dwells both upon published and unpublished papers, including private correspondences, to enucleate three fundamental questions, which are the cornerstones of Reichenbach’s caustic “attack” against Einstein’s and others’ attempts at unified field theory. In the first place, Giovanelli sheds light on the relation between a theory’s abstract geometrical structure (metric, affine connection) and the behaviour of physical probes (e.g. rods and clocks, free particles). In the second place, Giovanelli discusses whether one has to speak of a geometrization of physics or a physicalization of geometry. In the third place, he shows that Reichenbach drew attention to the interplay between geometrization and unification in light of field theory. Neuber’s chapter “Einstein’s Relativity from the Viewpoint of R. W. Sellars’s” will face a more recent and often underrated segment of the realist and operationalist interpretations of relativity. Indeed, he deals with Roy Wood Sellars, the father of Wilfrid Sellars, who was deeply committed to special relativity. To begin with, Neuber focuses on the analysis of Sellars’ The Philosophy of Physical Realism (1932) and the exposition of his operationalist account, according to which special relativity is but an “ars mesurandi”. However, Neuber also proves that such an operationalist overview instantiates a more “realist” and “absolute” standpoint about Minkowskian spacetime. Finally, Neuber compares Sellars’ vision with Bergon’s, as well as he addresses Sellars’ later approach (in the 1940s and the 1950s) in light of more recent endeavours aiming at “grounding” special relativistic kinematics. Napoli, Italy

Chiara Russo Krauss Luigi Laino

Acknowledgments

This book collects the papers presented during the Conference “Philosophers and Einstein’s Relativity”, organized by the University Federico II of Naples on 26– 28, May 2021 as part of the activities of the project SIR “Avenarius, Petzoldt and the Berlin Group”. The editors wish to thank all the lecturers and attendees to the conference, as well as the Italian Ministry of Research that financed the project.

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Contents

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2

3

4

(Mis-)Interpretations of the Theory of Relativity – Considerations on How They Arise and How to Analyze Them . . . . . . . . Klaus Hentschel

1

A Machian Interpretation of the Theory of Relativity? Joseph Petzoldt’s Reading of Einstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiara Russo Krauss

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The End of Matter? On the Early Reception of Relativity in neo-Kantian Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paolo Pecere

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Cassirer and Klein on the Geometrical Foundations of Relativistic Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Francesca Biagioli

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Natorp, Cassirer and the Influence of Relativity Theory on Neo-Kantian Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Luigi Laino

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Coordination, Geometrization, Unification: An Overview of the Reichenbach–Einstein Debate on the Unified Field Theory Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Marco Giovanelli

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Special Relativity from the Viewpoint of R. W. Sellars’ The Philosophy of Physical Realism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Matthias Neuber

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Contributors

Francesca Biagioli University of Turin, Turin, Italy Marco Giovanelli University of Turin, Turin, Italy Klaus Hentschel University of Stuttgart, Stuttgart, Germany Luigi Laino University of Naples Federico II, Naples, Italy Matthias Neuber University of Mainz, Mainz, Germany Paolo Pecere University Roma Tre, Rome, Italy Chiara Russo Krauss University of Naples Federico II, Naples, Italy

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Chapter 1

(Mis-)Interpretations of the Theory of Relativity – Considerations on How They Arise and How to Analyze Them Klaus Hentschel

Abstract During Einstein’s lifetime, the special and general theories of relativity were quite frequently interpreted by philosophers. Most of these interpretations actually were misinterpretations. Even today interpretative statements about relativity theory are often false or highly misleading. Why is this so? In my Ph.D. dissertation (Hentschel 1990a), I analyzed (mis)interpretations by 10 different philosophical schools active in the early twentieth century which widely differed in their approaches, emphasis and blind spots. Many of these interpreters – including philosophers of high standing such as Ernst Cassirer, Moritz Schlick or Joseph Petzoldt – had studied the theory intensely and many even had close contact with Einstein himself or with one of the members of his “protective belt” of close friends and allies. Rather than declaring all of these (mis)interpreters as either luminaries or idiots (which would be implausible, if not downright silly), I show structurally how these misunderstandings arose and why they were kind of unavoidable, even for highly qualified and often well-informed interpreters. More popular texts about relativity theory were often second-order interpretations of these first-order accounts, thus multiplying the first-order errors of misinterpretations. I will give a few characteristic examples but my focus will rather be on structural characteristics of these misinterpretations. I will discuss how to analyze them historically by means of interpretational frames. A link will also be made to Ludwik Fleck’s thesis that socially and cognitively stabilized “thought collectives” (“Denkkollektive”) exert strong constraints on human thinking and interpretation (“Denkzwang”). My concept of interpretational frames is one method to formalize and analyze the complex interrelations between different assumptions and inferences within such a frame of thinking (“Denkstil”). Semantic frames and word clouds are also discussed as alternative approaches but both are discarded as unsuitable for the purpose of reconstructing interpretative frames.

K. Hentschel () Section for History of Science & Technology, History Department, University of Stuttgart, Stuttgart, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Russo Krauss, L. Laino (eds.), Philosophers and Einstein’s Relativity, Boston Studies in the Philosophy and History of Science 342, https://doi.org/10.1007/978-3-031-36498-3_1

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K. Hentschel

Keywords Philosophical interpretation · Misinterpretation · Interpretational frame · Denkstil · Style of thought · Denkzwang · Projection · Preconditions & criteria for good interpretations · Perspectivism · Semantic frame · Taxonomies · Incommensurability · Talking past each other · Word-cloud · Adler · Besso · Bridgman · Cassirer · Einstein · Feyerabend · Fleck · Gadamer · Giere · Kraus · Kuhn · Mach · Massimi · Meyerson · Metz · Petzoldt · Reichenbach · Schlick · Solovine

1.1 Motivation for Studying Misinterpretations Interpretative texts on Einstein’s theory of relativity often contain errors, gross oversimplifications, misinterpretations and incorrect statements about the theories of relativity. Until 1919, the reception of relativity theory was concentrated to a few interested physicists, mathematicians, and philosophers with special interest and often former training in physics or other sciences; but even in these cases misunderstandings occurred.1 This intensified when Einstein suddenly became a world celebrity after the announcement of the first results of light-deflection observations by Eddington and Crommelin in late 1919.2 Philosophers of various philosophical schools, scientists with various epistemic, metaphysical, and normative opinions and later even journalists all contributed to a flood of ‘interpretations’ of the special and general theories of relativity full of oversimplifications, misunderstandings, and incorrect claims – therefore clear examples of misinterpretations. It would be easy to brush all of these publications aside as worthless rubbish, or to ignore them as ‘paper tigers’ – effigies to burn.3 But then one misses the chance to gain better insight into one of the most fundamental (and irritatingly frequent) processes of human thought, i.e., misunderstanding and misinterpretation. My claim is that just these systematic misinterpretations of Einstein’s theories can serve as a clue to a better understanding of the general cognitive process by which such misunderstanding arises. This phenomenon is by no means limited to philosophical interpretations of theories. It also occurs in popular and journalistic accounts allegedly only following common sense logic without any philosophical pretention. But its delineation and impact is clearer and thus easier to analyze in the detailed efforts at philosophical interpretation upon which we shall now concentrate.

1 On

the early (partly philosophical) reception of relativity theory see, e.g., Goldberg (1984): parts II-III, Maiocchi (1985), Glick (ed.) (1987), Hentschel (1990a, b, c), Biezunski (1992), Ryckman (2005) and Sanchez-Ron (2012). 2 See my book review of three of the most important recent historical studies on these light deflection expeditions and their context: Hentschel (2020). 3 As some so-called ,scientific biographies” of Einstein such as Abraham Pais (1982) do.

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1.2 Methodological Background In 1989, I submitted my doctoral dissertation on philosophical (mis) interpretations of Einstein’s special and general theories of relativity to the History of Science Department at the University of Hamburg; in 1990 it was published in the series Science Networks with Birkhäuser in Basel.4 A brief English summary appeared in PSA 1990 II, but the book is in German and hence not that well-known in the English-speaking world. My Sources Were (i) approximately 2500 contemporary published interpretative texts about the theories of relativity written by scientists, philosophers and laymen (ii) collections of unpublished documents preserved in the estates of physicists of that time, most notably the from the Einstein Archive at the Collected Papers of Albert Einstein, which contains copies of hundreds of letters by philosophers asking Einstein to explain features of his theories. They include Bergson, Bridgman, Cassirer, Metz, Meyerson, Petzoldt, Reichenbach, Schlick among others (still unpublished then in the 1980s but now readily available in the 15 published edited volumes of the CPAE as well as online at the site http://www. albert-einstein.org/ hosted by the Hebrew University of Jerusalem), and (iii) other unpublished materials in the estates of various philosophers, among these Mach (DMM Munich), Reichenbach and Carnap (Pittsburgh), Schlick (Amsterdam), Bavink (Bielefeld), Petzoldt (TU Berlin), or Friedrich Adler (Vienna).

1.3 What Can We Learn from These Sources? First of all, obviously different philosophers differ in their emphasis on their topic selection. Whereas the pioneer of operationalism, Percy W. Bridgman, focused on Einstein’s provocative statements about the relativization of space and time measurement in Einstein’s special theory of relativity, the critical rationalist Emile Meyerson rather indulged in an analysis of the theoretical unification achieved in Einstein’s general theory of relativity and gravitation. Hans Vaihinger and his fictionalistic followers saw in relativity a whole bunch of fictions, reinterpreting for instance Einstein’s axiom of the constancy of the velocity of light c as a counterfactual assumption (as if c were constant). And adherents of Husserl’s phenomenology sought to reconceptualize Einstein’s relativization of space and time as a mathematical expression of local space and time sensations.5 4 The book is freely available online for download at https://doi.org/10.18419/opus-7182 and at https://elib.uni-stuttgart.de/bitstream/11682/7199/1/hen47.pdf and via research gate; 5 See Hentschel (1990a, b, c) on Bridgman’s operationalism (pp. 425–440), Meyerson’s critical rationalism (pp. 456–472), Vaihinger’s fictionalism (pp. 276–292), and Husserl’s phenomenology (pp. 254–275).

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K. Hentschel

These four handpicked examples already show the breadth and variety of interpretations. Only interpretations of quantum mechanics show a similarly broad spectrum of approaches,6 but the theory of relativity is probably unique in the breadth and intensity of the ensuing debates.7 A comparative view of all of these philosophical interpretations furthermore shows the different strategies used by each philosopher in attempting to understand Einstein’s theories or a particular aspect of them. Henri Bergson myopically focused on the relativity of time and tried to introduce a categorical distinction between physical and physiological time in order to immunize the human experience of time from Einstein’s findings about time dilation of moving objects;8 whereas Hans Reichenbach systematically analyzed a broad array of aspects of both the special and general theories of relativity. Quite a few of these interpreters started to correspond with Einstein in order to discuss aspects of the theory with its originator – that is, until Einstein became tired of such discussions in the early 1920s and withdrew.9 A few even attended Einstein’s lectures or talks or tried to meet him during conferences etc.10 These exchanges of letters between Einstein and his circle of defenders exhibit an intense struggle to gain closer insights into science beyond the traditional realm of philosophy which was largely omitted or hidden in their later published writings. This internal dynamic and the genesis of various interpretations was a key topic in my book from 1990. This paper rather focusses on an analysis of the definite interpretations finally reached by these interpreters. Despite this restriction, there is still a danger of getting lost in the thicket of details and in the maze of arguments. Therefore, my book (and even more so this brief paper) looks for clusters of interpreters, for repeating patterns exhibited by more than one text and more than one philosopher. Here, I will focus on the interpretation of Einstein’s theory by Machians and on my methodology employed to analyze interpretations of a physical theory. As shown in the next section, these Machian interpreters got several features right and even claimed that Einstein was just fulfilling the intellectual demands of their spiritual forebearer Ernst Mach (1838–1916). But they also erred fundamentally in various aspects of their interpretation, even claiming that not they,

6 My

colleagues Olival Freire and Olivier Darrigol are currently editing a volume of studies on the multifarious interpretations of quantum mechanics forthcoming at Oxford Univ. Press in 2022. 7 For exemplary studies on the broad reception of relativity theory in various nations see the texts listed in footnote 1. Graham (1972) deals with the reception of both relativity and quantum theory in the former Soviet Union. 8 See Bergson (1922), (1924) and (1972) on “les temps fictifs et le temps réel”. 9 See, e.g., Albert Einstein to Elsa Einstein, May 19, 1920 (in Engl. transl. by Ann M. Hentschel): “I have to study Cassirer’s manuscript, which is less amusing. These philosophers are peculiar birds.” below on the consequences of this withdrawal for the dynamics of the ensuing debates, which were then led by a belt of self-declared Einstein defenders. 10 Reichenbach heard Einstein lectures in the winter term of 1917/18, and Petzoldt invited Einstein to discussions of the Berliner Gesellschaft für wissenschaftliche Philosophie which he had founded in 1912 and directed until 1921. See Hentschel (1990c) and Danneberg et al. (eds.) 1994.

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but Einstein must have made a mistake in the points under critical discussion. The same could be shown of most other participating philosophical schools of thought. Were these only individual choices of topic, skewed or idiosyncratic interpretations, personal mistakes or vagaries of dogmatism? No! Surprisingly many parallels in the writings by different persons emerged, esp. when belonging to the same school of philosophical thought. My suspicion arose that this was rather a kind of Denkzwang (in the sense of Ludwik Fleck11 ), a quasi-automatic interpretational move induced by a similar cognitive and interpretative background shared by all of these interpreters. Members of each philosophical school form a thought collective (Denkkollektiv) characterized by a limited number of members,12 an intense exchange of ideas, intense internal communication and similar attitudes and goals. This shared intellectual framework in which these representatives of one such philosophical school operate thus channels their thought: It forms a kind of thought milieu. The result is a Denkzwang, which explains why we find so many clusters of surprisingly similar interpretations, so many patterns of similar errors and misinterpretations, so similar selections of hot topics and blind spots overlooked by all members of this group. Therefore, for my kind of approach, it is not the individuals but the philosophical schools to which they belong that form the unit of study. Our focus is thus not on individuals but on schools of thought. As Fig. 1.1 shows, during Einstein’s lifetime a polyphony of interpretations of relativity already existed.13 Various philosophical schools discussed SRT and GTR selectively, picking out and emphasizing very different parts, downplaying or ignoring others. Obsession with some topics, as well as a kind of specific blindness toward other aspects of the theory of relativity is a common feature of all contemporary interpretations during Einstein’s lifetime.14 Several groups only dealt with very few isolated topics from both theories, otherwise showing no interest in their context or overall interconnections. Only few philosophical schools were ambitious enough to try to incorporate substantial parts of the theories into their interpretations (Reichenbach, Schlick and the logical empiricists, Petzoldt and a few Machians, Cassirer among the neo-Kantians). Yet others did not confine themselves to this restricted role of an interpreter, but openly refuted parts of both theories as being in conflict with their general tenets (double arrows in the previous figure mark such strong disagreements).

11 See

Fleck (1935/80, 1979, 2011) and Cohen and Schnelle (ed.) 1986.

12 See Collins (1998): 30, 54ff., 81f. on the “law of small numbers”, limiting the overall size of these

thought collectives as well as the number of people in their top level of intellectual forebearers and leaders. 13 On the pattern of competing philosophers or philosophical schools of comparable stature and its positive impact on philosophical creativity, see Collins (1998): 6 and 73: “The intellectual world at its most intense has the structure of contending groups meshing together into a conflictual supercommunity.” 14 For interesting examples of such specific blindness induced by training in the style of thinking of a specific microbiological thought collective, see Fleck (1935/80) chapters 1 and 3, and Fleck (2011): 213f.

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Fig. 1.1 Application of relativity theory (RT) to the ‘interpretational frames’ of some schools of philosophers (not comprehensive). (From Hentschel 1990b, PSA II: 170)

Of course, sometimes one can find psychological reasons behind these different reactions to a compelling scientific issue, stemming from the individual personalities of the interpreters (e.g., an arrogant Oskar Kraus versus a modest Ernst Cassirer). But mostly there are differences between interpretational frames (e.g., different models about the relationship between physics and philosophy) which ultimately account for these astonishing variations. Because all interpreters of one philosophical school belong to the same thought collective, they will all react similarly on certain points, hence they will all tend to emphasize the same points, and heavily protest against others. Their reactions and responses to certain issues of interpretation were mostly totally uncoordinated and they often did not know that someone else of similar philosophical persuasion was also writing about the same issue at the same time. Even unbeknownst to each other, they put forward astonishingly similar interpretations. They were all subject to the same Denkzwang, induced by their interpretative frame. A comparative view on these interpretations helps to identify the repeating patterns. For instance: Any critique of scientific results was a no-go for logical empiricists; they felt that they were only legitimated to analyze them, to place them in a broader context and, perhaps, thereby to make them more comprehensible to the public. By contrast, a realm of fundamental issues existed for neo-Kantians that could only be approached using a transcendental form of philosophy, which sought to deduce a-priori patterns of the world excluding any possible interference with

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empirical issues. Since space and time were among these issues, a conflict with STR was to be expected; in fact it was inevitable. So, there is something tragic about this uncoordinated, yet strong parallel in these reactions, comparable to a pack of dogs, all trained individually and at different times, but all responding very similarly and quasi-automatically to a given stimulus. This comparison between a deep intellectual interpretation and a stimulated response might sound awkward and disrespectful, my point is not to compare high-brow philosophers with primitive dogs – I rather aim at the automaticity of the conditioned response. Philosophy is of course not merely routinized training, but a deep embeddedness into an interpretational frame triggering these strongly correlated responses. Our eleven philosophical schools are good examples of Fleck’s Denkkollektive. In Fleck’s (mostly medical) case studies these thought collectives were socially defined (via teacher-pupil relations, personal training, close cooperation or joint publication). In our case they are rather cognitively defined (via shared assumptions concerning the epistemic, ontological and methodological core of their tenets). Even people who had never met each other and never corresponded with each other might be members of the same thought collective. The reasons behind the deep agreement in their very fundamental epistemological, ontological and methodological core assumptions are varied: it can be actual teacher-pupil relations (such as, e.g., the critical realist Aloys Wenzl (1887–1967), who was a pupil of Erich Becher (1882–1929) whom he also succeeded as professor of philosophy at the University of Munich).15 But more frequently these interpreters rather read the same texts, pick the same role-model and are thus influenced by the same intellectuals. Often their shared intellectual roots are the originators of the philosophical school under discussion (e.g., Kant for the Neo-Kantians, or Mach and Avenarius for the phenomenalists).16 In the latter case, with two originators of a philosophical movement, the interpretations by those situated more on the Avenarius-side (such as Joseph Petzoldt) might differ slightly from those on the Mach-side (e.g., Rudolf Lämmel), but as long as the joint assumptions in the core of the interpretational frame are coincident, there will still be these parallels in their philosophical interpretations.17 Once a Denkstil (and in our case an interpretational frame) is firmly in place, it “automatically carries out the greatest part of our mental work for us”,18 as Fleck says, and we often even do not notice its effect on the channelling of our thought, on our selection of topics, on the emphases we place and judgments we make. Even though each thinker believes that he or she is singular and perhaps even idiosyncratic in his or her way of conceptualizing, thinking and judging, he or

15 On

Becher, Wenzl and the philosophical school of critical realism see Hentschel (1990a, b, c): 240–253. 16 For similar structures in the history of philosophical thought since antiquity see Collins (1998). 17 For a discussion of the subtle differences between Avenarius’s empiriocriticism and Mach’s phenomenalism and for a detailed discussion of Joseph Petzoldt’s interpretation of relativity theory (who was a pupil of Avenarius but became an ardent follower of Ernst Mach), see Dubislav (1929), Krauss (2019, 2021) and her contribution to this volume. 18 Fleck (1935/80, 1979: 84).

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she is socially embedded in an intellectual community and intellectually bound to a set of norms and internalized core assumptions which direct the thought.19 Two conclusions can be drawn: (i) we (as interpreters of interpreters) have to reconstruct these interpretational frames. So far, I seem to have been the only one using such frames and schemata in order to document and analyze these interpretations. The form of our accounts, of our own talks and publications on such interpretations has to change. What I propose is not just the production of lengthy texts with long quotes and a maze of complex arguments: I am looking for more diagrams or schemata of interpretational frames, not only in my own texts, but also in publications by others in the history and philosophy of science. Given these interpretational frames it is easily possible to understand the process of selection and omission of topics. It also becomes clear why schools have run into systematic conflict with certain tenets while chronically overemphasizing yet others. In a diagram, more general and more specific assumptions are distinguished and their interdependencies and interrelations are clarified at a glance. When I published my book in 1990, my hope was that others would take up this method – none have so far (as far as I see) – hence my main motivation to write this paper. I want to encourage other scholars to take up this suggestion to utilize schematic interpretational frames, perhaps to improve upon it, and to combine it with more modern means of textual analysis coming from linguistics and from the new field of Digital Humanities.

1.4 The Example of Phenomenalism (Mach School) So much about programmatics – now let us get more specific and take a look at one such diagram (Fig. 1.2): my analysis of basic tenets of the interpretation of the theory of relativity provided by representatives of the Mach school of phenomenalists, including interpreters such as Joseph Petzoldt, Friedrich Adler, Anton Lampa and Rudolf Lämmel.20

19 As the sociologist Collins (1998: 7) writes: “thinking consists in making “coalitions in the mind”,

internalized from social networks, motivated by the emotional energies of social interactions”, and – I would add – also by cognitive, epistemic, ontological and methodological core assumptions, some of which define interpretational frames. 20 See Hentschel (1990a, b, c): 390–424 and Hentschel (1991) particularly on Petzoldt; cf. Dubislav (1929) and the paper by Chiara Russo Krauss in this volume, who distinguishes more closely between Mach’s phenomenalism and Avenarius’s empiriocriticism. Petzoldt stood under the influence of both progenitors; Avenarius and Mach vigorously applauded each other’s writings, so it is questionable whether we should look at both groups as a single school of thought or as two such schools, certainly very closely related ones, but perhaps not identical.

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Fig. 1.2 Interpretation of Einstein’s theories of relativity by phenomenalists. (From Hentschel 1990b, PSA II: 172)

At the top level of each interpretational frame (as in Fig. 1.2), we find two or three core assumptions of the philosopheme. This core of each philosophy is not always easy to identify since only very few philosophers (such as Descartes or Spinoza) explicitly state that they base their philosophical credo on this or that assumption.21 In our case, all Machians share two such core assumptions: (i) the epistemological and ontological restriction to phenomena perceived or at least in principle perceivable by human observers (hence the term phenomenalism), and (ii) the methodological requirement of theory-instrumentalism, reducing theories to an aid in obtaining the most economic representation of nature and restricting science to a systematized aid for human orientation. At the middle level, we find medium-level assumptions such as a rejection of metaphysics, the credo of Denkökonomie (related to theory-instrumentalism above), or a belief in the relativity of all empirical reports to the observing subject (a derivative of epistemological phenomenalism). At the third and lowest level we see specific components of the Machian interpretation of the theory of relativity. These interpreted parts of the theory are

21 One

is reminded of Einstein’s letter to Solovine, May 7, 1952, in which he drew a diagram on how to obtain these top-level axioms, starting from a base of empirical facts and observations, publ. in Solovine (ed.) 1956: 120f.

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easily identified from the published interpretations, but in Fig. 1.2 only a few of the most important statements (recurring particularly often as chronic components of the Machian interpretation) are listed. In all cases, these arrows leading from the top to the bottom do not represent strict logical implications – they rather stand for likely inferences, for plausible connections, not about provable and incontrovertible implication. We are here speaking about philosophical interpretations, not of logical deductions. These interpretations do have some play, which also explains why certain interpreters could change their minds and re-interpret certain issues at the bottom line of the interpretational frame by drawing other connections. Interpreting a scientific theory (similar to interpreting a text) is no exact science but a hermeneutic exercise in establishing plausible connections and likely inferences. It is holistic and hence it is always important to keep the full interpretational frame in mind, not just one little part of it. In the case of the Machians, their interpretation had quasi-automatic implications: • An outright rejection of all “metaphysical concepts” such as Newton’s absolute space & time, aether, . . . • and a strong reinforcement of the relativity principle: for all Machians all motion is relative. All of this is consistent with Einstein’s own heuristics, motivation and understanding of his theory at the time. It is thus no surprise that the Machians, and especially Joseph Petzoldt, presented themselves as the one and only veritable interpreters of Einstein’s theory of relativity. Hadn’t Einstein himself written admiring letters to Ernst Mach after repeatedly reading the latter’s critical history of mechanics (publ. in 1883) in which Mach had severely criticized Newton’s mechanics and called for a new mechanics, obeying the principle of relativity in the most general sense?22 The same seems to be true of Mach’s theory instrumentalism. Didn’t Einstein reject the metaphysical concepts of absolute space and absolute time, and didn’t he ask for operational definitions of length and time measurement? Wasn’t Einstein’s theory of relativity the perfect embodiment of Mach’s epistemological phenomenalism, the last execution of Mach’s demands? So far, the Machian interpretation seems to be in total harmony with Einstein’s own position. Mach as well as Einstein had demanded a strict economy of thinking. Both appreciated a vivid interplay of experimentation and theory formation, theoretical unity, simplicity and elegance. Up to here, everything seemed to be fully concordant with the theory of relativity. Therefore, Machians often claimed to be the best, if not the only legitimate interpreters of the theory. These claims were supported by the fact that the young Einstein had been an ardent supporter of Mach’s critique of Newtonian mechanics and an avid reader of his publications. In the

22 On

the complex relation of Newton, Mach and Einstein see, e.g., Einstein (1916), Petzoldt (1921), Ray (1987), Wolters (2012), Hentschel (2019) and further references cited there.

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obituary of Einstein (1916) of Mach he even wrote: “I even believe that those who consider themselves to be opponents of Mach hardly realize how much Machian thought they have suckled in, so to speak, with their mother’s milk”, contending that Mach himself might just as well have found relativity theory, had he been younger at the time when was working on mechanics.23 But there is a flip-side to this coin: further quasi-automatic implications for the interpretation of all Machians, leading to strong conflicts between Einstein’s theory and its Machian reading. Mach’s epistemological relativism led to a categorical rejection of Einstein’s second axiom of special theory of relativity, i.e., to a rejection of the constancy of the velocity of light in vacuo. This, of course, was not consistent with Einstein’s special theory and with his search for relativistic invariants. In other words: Einstein’s and Mach’s relativity principle did not mean the same thing. Einstein’s relativity principle was (mis)interpreted by the Machians as demanding that everything be relative to the observer, whereas Einstein’s goal in construing the theory of relativity was quite the reverse, i.e., to find physical magnitudes independent of such motion. Einstein even played with the idea of calling his theory a “theory of absolutes” (“eine Absoluttheorie”).24 Mach’s theory-instrumentalism led to a rejection of Einstein’s search for covariance or realism. This was not consistent with Einstein’s special theory of relativity nor with his search for relativistic invariance. The experimentalist Mach would consider theory only as a useful tool in order to come up with a practical means of orientation carrying the least theoretical garbage,25 whereas the theoretician Einstein invested all his efforts into finding the most fundamental and unified theory possible. In all of these aspects, the Machian interpretation thus suddenly came into strong conflict with Einstein’s own position and those of other physicists practicing relativity since 1910 (Planck, Sommerfeld, von Laue, Langevin, etc.). A case in point is Friedrich Adler (1879–1960), whom Einstein knew since his student days in Zurich. Adler had even altruistically declined a chair in physics at the University of Prague in 1909 on Einstein’s behalf. Einstein, in turn, publicly defended him when Adler had assassinated the Austrian prime minister Graf Stürgkh in 1916 and was incarcerated until 1918. They were thus close friends, but nevertheless Adler was driven into strong opposition to so many points of relativity theory that he ran into open conflict with Einstein. In 1917, right after

23 See

Einstein (1916): ,Ich glaube sogar, dass diejenigen, welche sich für Gegner von Mach halten, kaum wissen, wieviel von Machscher Betrachtungsweise sie sozusagen mit der Muttermilch eingesogen haben.“ On the young Einstein and his relation to Mach see the classic paper by Holton (1968), Pyenson (1985) as well as Hentschel (2019) and further references given there. 24 This expression was first used by E. Kretschmann, Über den physikalischen Sinn der Relativitätspostulate, A. Einsteins neue und seine ursprüngliche Relativitätstheorie, Annalen der Physik 53 (1917): 576–614 and then rapidly used by Wilhelm Wien and many others, including Einstein himself. 25 Mach (1872): 46: ,Das Ziel der Naturwissenschaft ist der Zusammenhang der Erscheinungen. Die Theorien aber sind wie dürre Blätter, welche abfallen, wenn sie den Organismus der Wissenschaft eine Zeit lang in Athem gehalten haben.“

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reading the draft of Adler’s treatise on relativity theory, Einstein wrote to another personal fried, Michele Besso in Berne, that Adler would defend silly points and clear misinterpretations of relativity theory with the cock-sureness of a prophet, leaving him at a loss about how to respond: “Adler is riding the Machian nag to exhaustion.”26 In his reply, Besso reminded Einstein, that this Machian nag “had after all safely guided Einstein through the Hades of relativity, and who knows whether he won’t also carry the rider Don Quixote de la Einsteina through those eval quanta”.27 Whereupon Einstein clarified his point by stating that he wasn’t complaining about the Machian steed; however, it would “couldn’t bring any viable insight, but only eradicate noxious vermin.” And if Besso would have had the privilege of reading Adler’s long and tedious treatise on relativity theory, he would understand perfectly well Einstein’s metaphor of Adler riding his Machian hack to death.28 Similar tales could be told of practically all the other philosophical interpretations of relativity theory. They had their (greater or lesser) strengths, but also their weaknesses. They brilliantly illuminated certain aspects of the theory whereas they miserably failed with others. Furthermore, they all ignored major parts of the theory altogether and were thus far from a complete interpretation of the theory as formulated by Einstein and other physicists at the time. A dogmatic and unreflective insistence on all core assumptions of the given interpretational frame would easily lead into conflict with core tenets of Einstein’s special and general theories of relativity. This feature of philosophical interpretation is nothing unusual and indeed only to be expected. An interpretation is not a small-scale reproduction of the full theory, but an illumination of certain parts considered by the interpreter to be the most interesting, the most fascinating, or the most repellant – or at least ones noteworthy enough for comment. Gadamer’s concept of “prejudicial structure” (“Vorurteilsstruktur”) – which according to Gadamer’s hermeneutics one should be aware of when interpreting other people’s thought – is an informal and nonschematized precursor to our interpretational frames. According to Gadamer, “a person who is trying to understand a text is always performing an act of projec-

26 Letter

by Einstein to Michele Besso 29.4.1917, publ. in Speziali (ed.) 1972: 106 (“Ich erhielt gerade ein in den letzten Tagen fertiggestelltes Manuskript über Relativität von ihm [Adler], in dem er mit der Ueberzeugung des Propheten recht wertlose Spitzfindigkeiten überaus breit darlegt, sodass ich in peinlicher Verlegenheit darüber bin, was ich dazu sagen soll. Ich zerbreche mir unaufhöflich den Kopf darüber. Er reitet den Machschen Klepper bis zur Erschöpfung”); cf also CPAE vol. 8a (1998), Doc. 331. 27 Michele Besso to Albert Einstein, May 5, 1917, in Speziali (ed.) 1972: 110: ,Was das Mach’sche Rösslein betrifft, so wollen wir es nicht verschimpfen; hat es nicht die Höllenfahrt durch die Relativitäten betreut? Und wer weiss. ob er nicht auch noch bei den bösen Quanten den Reiter Dom Quixote de la Einsteina durchträgt.“ 28 Einstein to Besso, May 1917, in Speziali (ed.) 1972: 114: “Über das Mach’sche Rösslein schimpf ich nicht. Du weißt doch, wie ich darüber denke. Aber es kann nichts Lebendiges gebären, nur schädliches Gewürm ausrotten. Wenn Du A.s langes und breites Elaborat genossen hättest, würdest Du mein Bild vom zu Tode gerittenen Klepper ohne Weiteres begreifen.“

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tion”29 Interpretation is thus not a passive reception of given input but active work with the text or theory under study, bringing forth certain aspects, illuminating it in a new light and putting it into different contexts, suppressing or downplaying other aspects. Gadamer seems to have hoped that open discourse thematizing all presuppositions and frequent passage through the hermeneutic circle would help to dissolve these presupposed assumptions – I do not think so. I am afraid that the ubiquity of deep misunderstanding in philosophical debates (not only about relativity theory) proves me right. The interpretational focussedness of all interpreters, already pre-tuned to certain issues and blinded to others, also has severe consequences. Interpretational frames lead to a strong selectivity of which issues are interpreted, whereas other issues are ignored or neglected. Those aspects of a theory to be interpreted which are in obvious affinity to core tenets of the interpretational frame are prioritized. Scheme-relevant information is studied with attention whereas scheme-irrelevant information is ignored or overlooked. The full interpretational frame also raises certain expectations about which issues should be present in the theory under interpretation. For instance, all Kantians looked for features of a priori knowledge, since their interpretational frame demanded that such features should be present in any viable physical theory. The whole theory is “projected” onto the plane of understanding opened-up by the interpretational frame. As in perspectival optics, where each object point is linked to an image point by a ray of light, each interpretational point is linked to a selected issue in the theory under interpretation (cf. my Fig. 1.2 which schematizes this intuitive analogy of optical projection and philosophical interpretation).30 In some cases, parts of the theory under study are integrated into the interpretational frame which is thus modified or revised (as in the case of the few Neo-Kantians who were willing to execute revision-strategies rather than immunization of their philosopheme). Furthermore, each interpreter claims his or her interpretation to be the best, perhaps even the only permissible one. Thus, different interpreters irretrievably had to get embroiled in serious squabbles about who is right or wrong, and who is the ‘proper‘ interpreter of Einstein‘s theory of relativity. In their opinion, only one of them could be it,31 but all of them were wholly convinced of their own line of (allegedly totally adequate) interpretation.32 Cock-fights among the major Einstein experts resulted about who was the legitimate spokesman. One indicator 29 Gadamer

(1960/85), part two, II: 236 never formalized or schematized this insight. On the classical hermeneutic stance with regard to interpretations, see Gadamer (1960/85, 1974, 1995) and the extensive literature on hermeneutics. A good survey of this in English can be found in the Stanford Encyclopedia of Philosophy, online at https://plato.stanford.edu/entries/hermeneutics/, consecutively written by Theodore George (2020), best in the third and last version. 30 The same metaphor is also used by Gadamer (1960/85), Giere (2006, 2009) and Brown (2009): 215. Hasok Chang (in Massimi and McCoy, ed. (2019): 21) warns his readers about “the very seductive and deeply misleading aspect” of the two metaphors of “projection” and “perspective”. 31 or none of them, but their own self-esteem prevented them from believing that! 32 This, again, is a feature well-known from the history and sociology of philosophy, with endemic and endless quarrels about who holds the ‘right’ interpretation of Plato, Kant, Hegel, etc.: see Collins (1998).

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of this would be whom Einstein applauded, whom he mentioned and recommended in his own publication: this guy (women were a rare species among them) would be it.33 Well, this shifted over time, from Mach and Petzoldt (in the early 1910s) to Schlick (between 1917 and 1920) to Meyerson (in 1925).34 At any rate, a “belt of defenders” around Einstein‘s theory of relativity emerged, including some of the best-known philosophers of science (Reichenbach, Schlick) and a few physicists with some interest in philosophy.35 As Randall Collins said with regard to the history of philosophy in general: “Philosophers of greatest repute tend to be personal rivals representing conflicting schools of thought for their generation.”36 The same happened in our case; so cases of fruitful exchange of ideas between these various self-proclaimed defenders were rare, and polemical, quarrelsome debates about isolated interpretational issues between them, as well as between themselves and the people outside who were trying to make some sense of relativity were the norm. Of course, conflicting positions and competing interpretations might further the discussion about the theory of relativity; but in our case, it just resulted in a big heap of misunderstandings and halftruths amongst which the few serious, careful and reflected efforts at interpretation were hard for contemporaries to identify. Many objected to the Carnivalization of relativity theory (“Relativitätsrummel”) and suspected Jewish publicity propaganda behind the plot or wrongly assumed that Einstein himself was the mastermind, holding all the strings in this game.37 Einstein’s opponents started to launch heavy and persistent assaults against all kinds of aspects of the theory in the early 1920s and were often rebutted by one of those Einstein- defenders, since Einstein himself had withdrawn from these fruitless and otiose debates in the early 1920s and rarely ever took up discussion with philosophers again. This led such critics to speak of pro-Einstein-propaganda by self-nominated spokesmen pretending to be strictly following Einstein’s theory, but in fact defending their own, often highly controversial interpretations. It is worthwhile to study these debates with Ludwig Fleck’s thoughts about Denkstil as well as Thomas Kuhn’s and Paul Feyerabend’s claims about incommensurability in mind.38 Sure, they spoke the same language and all considered themselves to be philosophers or scientist-philosophers, but that did not suffice. Sharing a common Denkstil is a necessary precondition for unambiguous communication – only to the extent that persons share a joint Denkstil will they agree in their view of

33 For

a survey of the philosophical interpretations of relativity theory in chronological order, see the contribution by Don Howard in this volume. 34 Some traces of these shifting allegations are found in Einstein’s Autobiographical Notes (1949). 35 See Hentschel (1990a, b, c): 163–195 and Hentschel (2006). 36 Collins (1998): 76. 37 For a survey of the front of often right-wing and anti-Semitic opponents of relativity theory, see Hentschel (1990a, b, c): 131–149 and Wazeck (2014) and further references given there. 38 See Feyerabend (1962), Kuhn (1962), (2000), Giere (2016) and Oberheim (2018) for further references.

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Fig. 1.3 Overview of the contemporary disputes about relativity theory and about its most important defenders, forming a kind of protective belt around Einstein, who himself retreated from these heated, but ultimately pointless controversies. (From Hentschel 1990b, PSA II: 175)

the world (and in their philosophical interpretation of physical theories); whereas persons brought up in different styles of thought will fail to communicate without friction. They will rather consider the other side to living in a fanciful world of fantasy.39 Hence many people were talking past each other in Einstein’s day (Fig. 1.3). Their philosophical misunderstanding about certain features of relativity

39 Fleck (1935/80, 1979: 109–11, 126f). and Patrick A. Heelan in Cohen and Schnelle (eds.) (1986).

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theory was aggravated by misunderstandings of these interpretations by groups of interpreters with totally different interpretational frames. So all of these frames were not being projected onto relativity theory anymore, but onto yet other frames of interpretation. Second- and third-order effects occurred in these debates about the theory of relativity, its interpretations, along with the interpretations and critiques thereof.40

1.5 A Clash of Styles of Thought and Interpretational Frames in These Debates Let us return once more to the characterization of thought collectives in the sense of Ludwik Fleck. In one of his not so well-known papers (originally published in Polish in 1936 and translated into German and English as late as 1979 and 1983), Fleck spoke of the “collective mental differentiation of men”: people exist who can communicate with each other, i.e., who think somehow similarly, belong, so to say, to the same thought-group, and people exist who are completely unable to understand each other and to communicate with each other, as if they belong to different thought-groups ( thought-collectives). [ . . . ] They will talk next to one another, but not to one another: they belong to different thought-collectives, they have other thought-styles. What, for one of them, is important, even essential, is for another a side issue, not worth discussing. What is obvious for one, is nonsensical for the other. What is truth (or ‘lofty truth’) for one of them, is a ‘base invention’ (or naive illusion) for another. Even after a few sentences there appears to be a specific feeling of strangeness, which signals the divergence of thought-styles, just as in other cases one experiences, even after a few sentences, a specific mental solidarity with our interlocutor, which proves an affiliation with the identical thought-collective.41

Directly after this passage, Fleck illustrates his claims with an interesting comparison of the writings of Bergson and Maxwell on motion. The result of his analysis of various longer quotes from both of their writings on what at first appears to be the same topic was the following: The two men would be unable to communicate with each other as regards motion: Bergson looks for the ‘absolute’, and believes that the ideal of cognition is the experience ‘from inside’. Maxwell looks for relations to and connections with the surroundings, and his basis of cognition is a relativism. According to Bergson, Maxwell studies the substitute for motion, while according to Maxwell Bergson studies illusions without any substantial content. Bergson finds fault with Maxwell, that he does not study at all ‘motion as such’, but only its symptoms for the sake of the configuration of the system. Maxwell finds fault

40 For

an exemplary analysis of one of these discussions between the critical realist Oskar Kraus, the logical empiricist Philipp Frank and the Neo-Kantian Benno Urbach, see Hentschel (1990a, b, c): 541–549. 41 Ludwik Fleck 2011, in German transl. by Werner and Zittel (Ed.): 263, 265f., originally published in Polish in 1936, transl. into English by Cohen and Schnelle (eds.) (1986): 81.

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with Bergson, that to experience motion does not mean to get to know it scientifically, but, on the contrary, it often makes its cognition impossible. It becomes evident that words have different meanings for Bergson and for Maxwell: the ‘motion’ of Bergson is something different from Maxwell’s ‘motion’, just as the same words ‘to get to know’ have different meanings for both philosophers. At bottom, almost all words have different meanings for them: it is not as if the word of one of them meant a thing that the other would give a different name, but that a thing to which one of them gave a certain name does not exist at all for the other one. That is why it is impossible to translate exactly the utterances of one of them into the language of the other one. Bergson’s motion, the motion in itself, absolute motion, does not exist at all for Maxwell; there exists no word to express it, nor is there a need for it, the same applies also to the ‘perception from inside’ of Bergson. In general, this philosopher has a much richer language, while the physicist limits very markedly the store of his words, and he does it on the basis of the specific tradition of ‘scientificality’: a certain discipline of thought, produced by the history of science, makes him give up some words as useless. The philosopher is not bound by this discipline, but he is bound by the specific tradition of philosophers who fundamentally do not give up any notions which remain from any period of thought.42

Very much the same kind of analysis could also be given in a comparison of Bergson’s and Einstein’s writings about time – and in fact I have done so in my book from 1990.43 This remote distance between the style of thought of Bergson’s intuitive “philosophy of life”, and Einstein’s physico-mathematical argumentation in theoretical physics necessarily produced the enormous gaps in mutual understanding. Hence Bergson’s efforts at interpreting Einstein’s theory of relativity failed miserably. I say so despite the heroic efforts of various later philosophers and historians at a rapprochement between the two thinkers. Not only Bergson’s and Einstein’s attempted dialogue during a meeting of the Société française de philosophie in Paris in 1922 and their later efforts to establish an international cooperation in the League of Nations failed, but Bergson’s efforts since 1921 to interpret Einstein’s theory likewise was an “experiment that failed”.44 Thus it was no surprise that their repeated volleys of arguments turned into an acrimonious debate. It is not that they did not show mutual interest in each other’s work from the start, but the philosopher Bergson simply was not willing to learn from the physicist and to revise some of the deeply embedded core-assumptions of his interpretational frame about time and life, reality and measurement. These representatives of [different] thought-collectives thus cannot establish understanding among themselves, although within the range of their collectives they not only understand each other, but even extend their systems of views in co-operation with other members of the group. Their thought-style is not an individual peculiarity, but a group one: it is based on a certain education and training and on a certain defined historical tradition. Thus, one should speak about different philosophical, scientific and mystical thought-styles. Each of these styles had passed through specific historical evolution, and each occupies a specific place

42 Ludwik

Fleck (2011), originally published in Polish in 1936, transl. into English by Cohen and Schnelle (eds.) (1986): 83. 43 For details about Bergson’s misinterpretation, see Hentschel (1990a, b, c): 441–455. 44 To borrow an article title by Jimena Canales (2005) whose book on Bergson and Einstein, Canales (2015), is an example of such a failed attempt at reconciliation.

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K. Hentschel in the mental life of mankind. We have quite a few such thought-collectives which act as carriers of more or less distinct thought-styles. They are produced by various separate forms of collective thinking, e.g., certain disciplines such as physics, philology, economics, or the knowledge of some practical professions such as craft, commerce, or next the knowledge of religious, ethnographic, political and other communities, the philosophical systems of definite schools, the so-called common-sense philosophy of life, etc. Some thought-styles approach one another, e.g., the physical and biological styles, others are more distant from one another, e.g., physical and philological, and finally some are as distant from one another as, e.g., the physical and the mystical. One can therefore speak of separate styles and of varieties of styles, and, similarly, of kindred and distant thought-collectives.45

In the above quote Fleck only distinguishes thought-styles among various disciplines. However, in other passages he says that the typical size of thoughtcollectives is only a few dozens – often only the experimentalists in a single lab or the members of a philosophy department in a specific town, or some other closely knit group living in a specific milieu and sharing a lot of intellectual, social and normative preconditions in their work. Hence in Einstein’s day we do not have one philosophical thought style but plenty of them, reflecting the existence of various rivaling philosophical schools. These philosophical schools were still relatively close institutionally (mostly grouped around professors of philosophy teaching at universities) and linguistically (mostly based in Germany, France, Great Britain or the USA, with sufficient language capabilities to read each other’s texts and in many cases also aided by good translations). Nevertheless, mutual exchange was rare and distrust, even disgust about one another’s approaches was widespread. How much the Machians and the logical empiricists hated the old-fashioned ontological and dispicably “metaphysical” talk of the more traditional philosophers! And how much the latter tried to prevent the former from gaining a foothold at German universities! As a rule, communication is only possible within one collective while within kindred collectives it is feasible only with some complexity: the inter-group exchange of ideas is always connected with a more or less marked modification of the ideas. When passing from one group to another, words change their meaning, the ideas obtain a different style colouring, the sentences receive another meaning, the opinions a new value. If the groups are considerably distant from each other, the exchange of thoughts can be completely impossible, and transformation of a thought consists, in such a case, in its complete destruction.46

No wonder that in the 1960s, Thomas S. Kuhn and Paul K. Feyerabend took up this line of thought and generalized it into their (controversial) talk about untranslatability and incommensurability.47

45 Ludwik

Fleck (2011), originally published in Polish in 1936, transl. into English by Cohen and Schnelle (ed.) (1986): 84. 46 Ludwik Fleck (2011), originally published in Polish in 1936, transl. into English by Cohen and Schnelle (ed.) (1986): 84f. 47 See Kuhn’s foreword to Fleck (1935/79), a source text which he had allegedly “simply forgotten” to mention in his famous book about the “Structure of Scientific Revolutions” in 1961. Compare the literature listed in Oberheim (2018) on this hotly disputed issue.

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1.6 Interpretational Frames and Perspectivism My book on contemporary interpretations of relativity theory was published in 1990 in the Series “Science Networks. Historical Studies”. It received wide notice among historians of science and physicists, was frequently reviewed, often recommended and was even awarded two academic prizes.48 In philosophy of science however, the reception was comparatively meagre, and only in German and Italian,49 perhaps due to the fact that the book was published in German and not translated. In 2000, the Minnesota-based philosopher of science Ronald Giere gave his first talk on the “perspectival nature of scientific observation”, and in 2006, he published his last book, Scientific Perspectivism (Univ. of Chicago Press). Quite a few historians and more philosophers of science then followed suit and also published on “perspectivism”, including Bas van Fraassen, Alexander Rüger, Margaret Morrison, Hasok Chang, Matthew J. Brown, Michela Massimi et al.50 Today a whole movement or strand of philosophy exists, called “Perspectivism”, sometimes also “perspectivalism”. William Wimsatt preferred to call his own position “multi-perspectival realism”, but they all uphold a view of scientific knowledge different from the standard (Nagelian) “view from nowhere”.51 My prior approach fits this label ‘perspectivism’ perfectly, I think. This also applies with respect to what Michaela Massimi (2018a, b) calls ,perspectival realism“ and to what Bas van Fraassen (2008, chap. 11) calls “empiricist structuralism”. Despite the polyphony of interpretations, they are all about the same theory, each of them tells us something about the theory. They all attempt to ,get things right“ – even though they often fail due to interpretational blindness. Hence we have ,perspectival realism“ not only in epistemology and philosophy„ but also in interpretations of theories in science. This is very important because it sets philosophical perspectivism apart from postmodern relativism and from an arbitrarization of science in which science is declared to be yet another mere expression of interest groups and power cliques, allegedly dominating discourse and crowding out minority views. Neither “scientific perspectivism” in the sense of Giere (2006) nor interpretational perspectivism in my sense degenerates into a

48 One

Ph.D. prize from the Hamburger Wissenschaftliche Stiftung, and a Heinz Maier-Leibnitz Förderpreis from the German Federal Ministry of Science and Education. Book reviews appeared in Isis 84,2 (1993): 404–5, British Journal for the History of Science 28 (1995): 482–3, Archives Internationales d’Histoire des Sciences 1996: 83–4, Gesnerus 49 (1992): 99–101, Annals of Science 49 (1992): 577–583, Foundations of Physics 22,12 (1992): 1517–1520, Physik in unserer Zeit 23,4: 183, Physikalische Blätter 48,2 (1992): 123–4, and in the Berlin newspaper: Der Morgen, April 4, 1991: 21. 49 See the essay review by Ferrari (1992). 50 See, e.g., Brown (2009), Morrison (2011), Massimi (2012), (2018a, b), Rueger (2020), as well as van Fraassen (2008), Wimsatt (2007). 51 The introduction to Massimi and McCoy (2019) offers a good survey of the movement in its “kaleidoscopic” character and a clear positioning of “scientific perspectivism” with respect to pragmatism, pluralism and realism.

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silly relativism. Actually, the same unfair charge was also brought against Einstein and his defenders. They were denounced as relativists, even as propagandists and as a Jewish clique or Jewishly-infiltrated interest group.52 If Einstein’s critics could have recognized and analyzed the interpretative frames behind the defensive and interpretative statements by members of the protective belt around Einstein, they would have realized that these defenders certainly did not all share the same opinion and they would then have better understood why each of them reached their separate conclusions in these debates. They might even have understood the deeper reason for their own failure to properly understand key facets of Einstein’s theory of relativity. According to Gadamer, a central tenet of all adequate interpretations would have been this ability and willingness to reflect about one’s own assumptions (here one’s own interpretative frame). But of course, in the heat of the politically and antisemitically loaded debates of the 1920s, such hermeneutic reflexiveness had no chance. Up to this point, Gadamer’s hermeneutics, Giere’s perspectivism and my approach using interpretative frames can be brought into accord. (i) Human and scientific observation, scientific theories as well as philosophical interpretations are all perspectival, (ii) all of these “perspectives” are asymmetric interactions between human (i.e., biological, cognitive, and social) factors and the world. (iii) all of these perspectives are partial and of limited accuracy. (iv) All of the above “perspectives” are neither objectively correct nor uniquely veridical – they all yield representations, i.e., limited views of their object.53 Hence, “perspectivism entails pluralism” and puts “emphasis on the plurality of epistemic goals”54 of different practicioners of science – and I would add – as well as of philosophical interpretations thereof. But – pace Giere, Brown et al. – there also remain crucial differences between perspectival seeing and interpretational perspectives: When we see a tree from one side, or the lit side of the moon, we do not infer that this is all there is – we immediately know or assume that there is more to see, and try hard to change our perspective to get a better idea of the full object. Consider the analogy with cartography: All maps always have to choose a vantage point, a projection technique to map a 3D spherical world onto a flat 2D surface, a scale and a resolution, a focus (e.g., on geographical topology, or on routes of transportation, or on industries and settlements, etc.). But all of us will still automatically presuppose that all of these different maps show the same landscape or city or world in which we live. In contrast to this, philosophical interpreters (and often “common sense”interpreters as well) immediately – but wrongly – assume that they have got it all 52 On

the genesis and dynamics of antisemitic arguments against Einstein and his theories, as well as against “Jewishly tainted” (allegedly “jüdisch versippte”) defenders of Einstein, see Hentschel (1990a, b, c): 131–149, Hentschel (ed.) (1996) and Wazeck (2014) as well as further primary and secondary sources cited there. 53 This list uses the brief summary which Brown (2009, 214) gave for Giere’s “scientific perspectivism” and extends it to philosophical interpretations which also yield such “perspectives” of the theory under interpretation. 54 Massimi and McCoy (2019): 3.

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(or at least the ‘essence’ of it). They forget their own interpretative frame, their philosophical filters through which they have studied their object. They all ignore or forget point (iv) of the above list. Giere’s lessons and mine are thus slightly different. Giere claims: • “Scientific Perspectivism argues that the acts of observing and theorizing are both perspectival,” I agree, and add: this is also true for the act of interpreting. • “and this nature makes scientific knowledge contingent, as Thomas Kuhn theorized 40 years ago.” I continue differently: this means that any interpretation has to be gauged against the background of the interpreter, hence we have to look for the interpretative frame. • “All models are perspectives on the target system.” However, scientific realism is not jeopardized since the target system is not shaped or construed by these perspectives – it is only described or interpreted in parts. No interpretation aims at giving a full and complete account. • “All of these different interpretations are interpretations of the same theory.” Yes, I admit this but their coverage, their mapping is strikingly different in breadth, quality and overall consistency. • Hence a pluralism about modeling is needed. Yes, but this does not mean that all interpretations are equally good or equally complete or equally sensible (see below for my list of criteria of evaluation). Hence, I plead for avoiding interpretational relativism. Talking about perspectivism and the inevitability of an interpretational frame naturally leads to the question of whether or not everything is relative (to the interpretational frame chosen by each interpreter). Yes, admittedly: different interpreters have different such frames, but each of them has one and cannot do without it. Perspectivism is thus unavoidable. Does this imply that all perspectives are equal? That all interpretations are equally valid? Or incomparable? I do not think so! Incommensurability does not imply incomparability.55 In order to avoid this relativistic impasse, we need criteria for quality assessment and for a fair comparison of different such interpretations. Here is my list of (straightforward) criteria for quality and comparison: • • • • • • •

Breadth Depth carefulness Naturalness Overall harmony and coherence Currentness (being up to date in your field of knowledge) Proper historical and philosophical contextualization

55 Oberheim

(2018):2 emphasizes that Kuhn himself always stressed “that incommensurability neither means nor implies incomparability”; cf., e.g., the intense discussion about this point, documented in Kuhn (2000).

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I think (or should I say I am under the illusion?) that all of these criteria are (or ought to be) uncontroversial. It should make a difference whether an interpreter is only picking one raisin off the cake and omitting all the rest, or whether he or she is really delving into it (even if the cake might suffer considerably more from the latter than from the former, i.e., interpreting more rather than less, of course, increases the chance of going amiss but so be it. Understanding the parts presupposes understanding the whole, and vice versa, understanding the whole presumes that one has a full view of all (or very many) of its parts. Since this seems to be a chicken-and-egg problem, the only way out of this dilemma is to constantly aim at a new and better understanding not only on the basis of already acquired knowledge and belief, but through “renewed interpretative attention to further possible meanings of those presuppositions which, sometimes tacitly, inform the understanding that we already have”.56 I would associate that with what I called “carefulness” above, but breadth also plays a role here since this hermeneutic circle will produce unwanted shortcuts if too little material from the theory under interpretation is taken into account. What is true for interpretative breadth is also true for depth. Bergson’s shallow (mis-)understanding of the relativization of time is simply not on the same intellectual level as Reichenbach’s or Schlick’s subtle interpretations thereof. Without sufficient depth of understanding, a philosophical interpretation remains shallow and pale. Therefore, Einstein shied away from discussing these obvious (mis)interpretations with their protagonists, be it the Machian Friedrich Adler or the philosopher of life Henri Bergson.57 In 1923, Einstein wrote to his friend Solovine that “Bergson, in his book on the theory of relativity, had made some serious blunders; may God forgive him”.58 It is also not surprising that the overwhelming majority of early efforts to understand and interpret Einstein’s theories is nowadays obliterated and forgotten. All of these hasty and sloppy interpretations simply left no traces, and a Whiggish historian might add: they were totally bogus to start with. If interpreters have to bend over backwards in order to come to grips with certain facets of relativity (like Petzoldt with regard to the constancy of the velocity of light in vacuo which he only accepted in his last publications after a lot of discussion with Einstein and relativists in Berlin), this simply uncovered a fundamental problem in his interpretative frame. The Machian interpretational frame had to be heavily reconceptualized in order to cope with this second axiom of the special theory of

56 In traditional hermeneutics, this leads to the idea of the “hermeneutical circle”; cf. George (2020)

sect. 1.3.and further references given there. Adler’s failed efforts at philosophical interpretation and physical critique of relativity theory see here note 26; on Einstein’s refusal to take up discussion with Bergson, cf. Canales (2005), (2015). On Einstein’s behalf, the former general and Meyersonian André Metz then heavily criticized Bergson in France and claimed him guilty of having transformed a beautiful child into a monster.” In the Spanish-speaking world, Masriera Rubio, a professor of physical chemistry in Barcelona, became a defender of Einstein 58 Einstein an Solovine, May 20, 1923, in Solovine 1956: “Bergson hat in seinem Buch schwere Böcke geschossen. Möge Gott ihm vergeben.” 57 On

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relativity, since it ran so much against the grain of complete relativization according to Mach’s phenomenalism. Hence, naturalness, coherence and overall harmony of the interpretation are also an obvious virtue that during Einstein’s lifetime only very few interpretations could claim for themselves. For Einstein himself, the “naturalness” of physical theories or “logical simplicity” of its axioms was also of key importance, and he himself always strove to optimize that.59 When assessing the relative merits of competing physical theories or when choosing which research path to follow, judgment about incommensurable qualities was needed. Einstein knew how difficult this was and how much of a nose for subtleties one needed, but very few philosophers of science were sensitive and sophisticated enough to appreciate this in their interpretations (Meyerson coming closest in this regard). Currentness could (and still can) only be achieved by interpreters willing to come into closer contact with the originators of the theory and its practitioners. Very few of the early philosophical interpreters of relativity theory had heard lectures by Albert Einstein (such as Petzoldt or Reichenbach in Berlin) or had worked on exercises in related topics during their physics education (such as Schlick who had been a Ph.D. student of Max Planck in Berlin before he switched to philosophy of science in Vienna). Typically we have a time-lag of roughly 5–10 years between the origin of a new physical theory and its philosophical interpretation (starting around 1915 for Einstein’s special theory of relativity (originating in 1905) and in 1920 for Einstein’s general theory of relativity, formulated in 1915).60 Proper historical and philosophical contextualization of interpretations only arose after Einstein’s death in 1955 in texts by Max Jammer, Adolf Grünbaum, John Norton or Michael Friedman, to name but a few of the later leading “histosophic” interpreters of Einstein’s theories during our times. But our “interpretational perspectivism” is very much in harmony with the historicity of science and scientific knowledge” which also adds constraints on the philosophical interpretations made.61 My own historico-philosophical comparison of interpretations of both theories of relativity during Einstein’s lifetime led to the result that among the contemporary interpretations of the theories of relativity, Hans Reichenbach fulfilled all preconditions and most of my criteria for a good interpretation. Moritz Schlick’s interpretation, preferred by Einstein between 1915 and 1925, was too cursory – his early violent death prevented his writing a more detailed account. Émile Meyerson’s interpretation, favored by the later Einstein, and André Metz’s later contributions in the spirit of Meyerson, worked as a sort of complementary interpretation focusing

59 See,

e.g., Einstein (1949) for reflective considerations about this lifelong goal which he thought to have met in relativity theory but never achieved in quantum theory or unified field theory. 60 For statistics on the reception of relativity theory see Eisenstaedt (1989), Hentschel (1990a, b, c): 67–73, Goenner (1992), (2017) and Hu (2005). For comparative studies of the reception in various national contexts, see here footnote 1. Faster media will maybe reduce, this time lag a bit in the future. 61 This proximity of perspectivism to pragmatism and to a historized view of science which goes hand in hand with an integrated history and philosophy of science is emphasized in H. Chang’s contribution to Massimi and McCory (2019).

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on points underestimated by the logical empiricists, such as theoretical unification and mathematical elegance. But in toto Meyerson’s and Metz’s interpretation was certainly less exhaustive and broad than Reichenbach’s. As we have seen in our case study above, Joseph Petzoldt, Friedrich Adler and other Machians over-emphasized the phenomenological aspects and misunderstood relativistic axiomatics. The general theory of relativity, in particular, doesn’t seem to have received an equally balanced treatment by contemporaries – its complexity was appreciated only later.62 I would like to end this paper along the lines of my book from 1990 – with a few recommendations for quality interpretations, since I assume that this proceedings volume about philosophers and relativity will be read by many young readers perhaps hoping or planning to interpret some recent scientific theory. This in and of itself might be a good idea, especially if a plethora of other such interpretations aren’t available yet. As the foregoing discussion has shown, there are various dangers and traps to fall into if one is not careful to keep in mind the following points: • Modesty and open-mindedness: it is always a good idea to be aware that one is trodding on foreign terrain. In fact, during Einstein’s lifetime very few philosophical schools excelled in that! Arrogance among philosophical chairholders abounded and was a major reason why new approaches in philosophy such as logical empiricism only succeeded later in exile. In the host countries of USA, Great Britain, Australia and New Zealand, the German and central European university autocrats were replaced by teams with flatter hierarchies and a collaborative spirit which allowed such non-standard strands in philosophy to grow. • Aim at a precise understanding of the contents, including mathematical formulars, intratheoretical derivations, the interplay of theory and experiment, the overall interconnections and contexts. The logical empiricists and Meyersonian rationalists performed best on this score, whereas the often boundless arrogance with which most other philosophers of Einstein’s time looked upon at themselves as the self-pronounced “kings” of science and humanities (Philosophie als Königsdisziplin oder Dachwissenschaft) and immediately assumed they knew it better, clearly worked against achieving this goal. No one less than the outstanding Australian philosopher of nature, John Passmore (1914–2004), admitted this much in his classic account of Hundred Years of Philosophy: “it must be confessed, professional philosophers have been intimidated by the mathematics into which philosophical physics so gratefully sink at crucial points of their reasoning; nor has the philosophical crudity of what they could understand led philosophers to expect any considerable degree of illumination from what passes their comprehension.”63 62 E.g.

in writings by Adolf Grünbaum, Michael Friedman or Chistopher Ray (to name just a few). (1966): 334. He admitted a few exceptions, though, including the Hegelian interpreter R.B. Haldane, as well as those interpreters who had transformed into philosophers after receiving a physical or mathematical training, such as Hans Reichenbach, Moritz Schlick, Alfred A. Robb or Alfred North Whitehead.

63 Passmore

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• In case of conflicts between one’s own interpretational frame and the theory under study: a willingness to revise conflicting (and perhaps incorrect) assumptions in one’s own point of view. Actually, very few members of any philosophical schools were willing to do the latter. Ernst Cassirer is one of the rare exceptions.64 What Hans-Georg Gadamer wrote in his classical text about a general hermeneutical understanding of texts can also be adapted to philosophical interpretations of theories: “Who-ever wants to understand [a text, or interpret a theory, for that matter] will not be allowed to rely on his own preconceptions [and on his interpretational frame] from the start in order to consistently and stubbornly think past the opinion of the text [or theory] under interpretation – until this misunderstanding becomes too obvious to ignore any further and topples the supposed meaning. Whoever wants to understand a text [or interpret a theory] will have to be ready to learn from it [and if necessary to revise his interpretational frame].”65

1.7 Summary and Considerations About Where to Go from Here I claim (i) a philosophical interpretation of a physical theory is an art, requiring – aside from knowledge – special skills which during Einstein’s lifetime only very few interpreters had. Psychologically crucial for a success are carefulness and longevity (most interpreters were (and still are) prefactory, fleeting hitand-run interpreters). A good interpreter should aim at breadth and depth of understanding; he or she should try to understand the naturalness and overall harmony of the theory to be interpreted and try to capture this in his interpretation thereof. Currentness (being up to date in the field of knowledge) as well as proper historical and philosophical contextualization are remote goals, usually only reached in later interpretations. (ii) all philosophical interpretations are perspective-dependent and insofar socially, historically and intellectually situated within particular contexts. This

64 Hence

in Hentschel (1990a, b, c): 196–239, I distinguished between immunization and revision strategies within (Neo)Kantian interpretations of relativity, with Ernst Cassirer, Josef Winternitz and Karl Bollert as main examples of the revisionists whereas dozens of other Neo-Kantians excelled in immunization strategies, usually of the type: . 65 Gadamer (1960/85): 253: “Wer verstehen will, wird sich von vornherein nicht der Zufälligkeit der eigenen Vormeinung überlassen dürfen, um an der Meinung des Textes so konsequent und hartnäckig wie möglich vorbeizuhören – bis etwa diese unüberhörbar wird und das vermeintliche Verständnis umstößt. Wer einen Text verstehen will, ist vielmehr bereit, sich von ihm etwas sagen zu lassen.“ Additions in square brackets are made by the author.

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philosophical perspectivism (as one might call this position) is in strict parallel to Ronald Giere’s claims about scientific perspectivism. What I have demonstrated in extenso for philosophical interpretations of relativity theory is structurally valid also of semi-popular interpretations of it (e.g., by Lenard and Stark in their ,Deutsche Physik“ – they also project Einstein‘s theory onto their self-declared common-sense plane of understanding); similarly so for many other interpretations given then and now. (iii) If what I claim is valid for interpretations of relativity theory, it should also work for the analysis of philosophical interpretations of other theories. Quantum mechanics is an interesting case since aside from relativity theory no other scientific theory of the twentieth and twenty-first century has been interpreted more often, and none more often (mis)understood.66 The same approach might work for an analysis of popularizations of theories. Popularizers also have their own frame of understanding even if it will be less categorical and harder to spell out. (iv) At any rate, working with interpretational frames clarifies the claims made as well as their interdependencies, and it explains why misunderstanding occurs so frequently, indeed compulsorily. As the pioneer of hermeneutics, Friedrich Schleiermacher (1768–1834), already knew: Misunderstanding is ubiquitous and it arises automatically – a proper understanding, however, must be actively sought for.67 As we saw above, this striving for proper interpretation comes at high cost. Rigidly clinging to interpretational frames also explains the astonishingly strong Denkzwang under which all proponents of philosophical schools stood when they started to interpret Einstein’s theories. One reason why the claims by Fleck about Denkzwang – and for that matter, mine about the effects of interpretational frames – are not noticed and implemented frequently enough, might be that we all believe in the illusion of a total freedom of thinking. In fact, our thinking is bound within the confines of these semantic and interpretational frames which we all use in order to understand the world as well as scientific theories trying to make sense of it. Philosophical interpretation is thus a second-order interpretation and the vicious and pointless debates between adherents of different schools of thought are actually a third-order effect of interpreters interpreting other interpreters. No wonder that endless debates, chronic misunderstandings and intellectual confusion are the result: an odd finding given the brilliant intellects of thinkers such as Cassirer, Schlick or Meyerson. Misunderstanding is part of the condition humaine. The clarification provided by interpretative frames is one means of coming to grips with this phenomenon.

66 See

footnote 5 above on Freire’s and Darrigol’s anthology of papers about these strongly differing interpretations. 67 Schleiermacher in his Hermeneutik (1838), § 15 & 16, Werke Abt. I, vol. 7: 20f., quoted in Gadamer (1960/85): 173.

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Fig. 1.4 Partial semantic frame for Maxwellian electrodynamic action dependent on the motion of either the magnet or the conductor. (From Andersen and Nersessian 2000, 238)

Are there other methods by which one might also achieve this goal? Of course, I have asked myself this question frequently, way back in the 1980s when I wrote my thesis as well as later, when new approaches and techniques were being put forward in history and philosophy of science by Ian Hacking, John Norton, Nancy Nersessian and other “histosophers”. One such technique is to reconstruct taxonomic trees by historians of science analyzing deep semantic and ontological changes during scientific revolutions; or lexical or semantic frames and schemata by linguists reconstructing human thought and language. One harmless example of this is in Fig. 1.4 showing such a partial semantic frame for Maxwell’s electrodynamics. More specifically, it delineates the interrelations between electrodynamic action, the states of motion or rest of a conductor, the aether and a magnet, and the ensuing formulas that have to be used in order to describe the various effects in action. Depending on whether the conductor or the magnet is at rest, two different formulas have to be used to describe the resulting electrodynamic action: In the one case the electromotive force (derivable directly from Maxwellian electrodynamics), in the other case, the so-called Lorentz force, postulated by Hendrik Antoon Lorentz (1853–1928) in 1895 in order to close a gap left by Maxwell for the case in which the conductor is moving and the magnet is at rest. In the context of Einstein’s special theory of relativity, this distinction between the two cases does not make sense anymore and the electrodynamic action was derivable by invariant means, independently of this classical distinction where “moving” means “moving with respect to the aether” and “at rest” is equivalent to “at rest with regard to the aether”. The complex semantic frame of Fig. 1.4 thus

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Fig. 1.5 Partial semantic frame for Einsteinian relativistic electrodynamics of moving frames of reference, magnet or the conductor. (From Andersen and Nersessian 2000, 238)

shrinks to a structurally different scheme in Einstein’s theory (Fig. 1.5) in which the aether does not appear any more. Analogously„ one can reconstruct such semantic and lexical frames for terms such as gravitation, mass, energy, space, time, motion, field, light, or for other core concepts of Einstein‘s theories of relativity. The insight gained by such semantic frames is considerable (which strengthens my call to revert more often to such schematic means of analysis). But it would be impossible to analyze a whole philosophical interpretation in this way. The label of the diagram already signals that this is but a “partial semantic frame”. It’s great for detailed analysis of specific points such as conceptual and taxonomic change, but not suitable for a holistic view of the full interpretation. Are there yet other ways to obtain an interpretational frame other than by careful study of the interpretation, followed by a reconstruction of the interpretative core? Are Word-Clouds an alternative? Not really: Word-Clouds are too coarse and unspecific. They do not yield representations of inferential structures in the frame, but they do offer a good overview of the interpretational content. Let us look at three examples (in Figs. 1.6, 1.7 and 1.8): a word-cloud generated from Joseph Petzoldt’s article on absolute and relative motion (1908), interpreting Ernst Mach and Ludwig Lange, contrasted against a word-cloud generated from his publication from 1912 on relativity theory in positivistic context and a from a paper on relative and absolute motion from 1920.68 All three word-clouds were generated with the program “word it out”, freely available at https://worditout.com/. All interpretationally irrelevant words (he, the, is, . . . ) were removed from the word list on which the word-cloud is based by KH in order to obtain useful results. 68 See

Petzoldt (1908, 1920) as well as Petzoldt (1921) and Graßhoff (ed.) (2006) for further comments and reprints of some of these earlier texts on the foundations of mechanics.

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Fig. 1.6 Word-Cloud for the article by Joseph Petzoldt (1908) on absolute and relative motion

Fig. 1.7 Word-Cloud for the article by Joseph Petzoldt (1912) on relativity theory in positivistic context

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Fig. 1.8 Word-Cloud for the article by Joseph Petzoldt (1920) on causality and the theory of relativity

The first word-cloud exemplifies the semantic and lexical context of discussions about foundations of classical mechanics as practiced in Mach’s Mechanics from 1883 and related papers by Carl Neumann, Ludwig Lange and others; the other two are clearly within the semantic universe of Einsteinian relativity, with keywords such as Einstein and Lorentz, velocity of light, coincidences and causality. So, lexical frames and semantic schemata might be a convenient way to clarify taxonomic interrelations as well as their (partly drastic) changes during scientific revolutions. Word-clouds or linguistic collocation analysis might be a quick and easy way to figure out where the emphasis of an interpretative text is. But they do not give the overall picture, nor do they make us understand the immense force of canalization of thought, the Denkzwang, which explains the chronic misunderstandings and misinterpretations of relativity theory by contemporaries. Interpretational frames are the best available means to obtain this goal and I would be happy if the method were taken up and refined by other scholars in science studies broadly conceived.

References Andersen, Hanne, and Nancy Nersessian. 2000. Nomic Concepts, Frames, and Conceptual Change. Philosophy of Science 67 (Suppl): 224–241. Bergson, Henri. 1922. Durée et simultaneité: a propos de la théorie d’Einstein, Paris; reprint in Bergson: Mélanges, 1972: 58–244.

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Chapter 2

A Machian Interpretation of the Theory of Relativity? Joseph Petzoldt’s Reading of Einstein Chiara Russo Krauss

Abstract Even though the relationship between Einstein and Mach is well studied, the literature on the topic often overlooks the fact that Mach never provided an interpretation for the theory of relativity. Rather, it was Mach’s pupil Joseph Petzoldt who published several works to prove that Machian philosophy provided the correct philosophical framework for the theory of relativity. The paper reconstructs Petzoldt’s philosophy and his interpretation of the theory of relativity, also showing how his conceptions sometimes differed from Mach’s. Thus, the study also explains why Einstein expressed a favorable opinion of Petzoldt’s work. On the one hand, the overall Machian tone of Petzoldt’s philosophy resonated with Einstein’s own Machian perspective. On the other hand, Petzoldt had developed the notion of Eindeutigkeit, which tended towards a stronger determinism (in contrast to Mach’s Humean conception of causality), and Einstein appreciated this deterministic version of Mach’s thought. Keywords Joseph Petzoldt · Ernst Mach · Albert Einstein · Special relativity · Eindeutigkeit · Causality · Determinism · Relativism · Perspectivism

2.1 Introduction When Einstein moved to Berlin in 1914, he was not yet the celebrity he would become after the announcement of the results of the 1919 solar eclipse observation, which confirmed the prediction of general relativity about the curvature of light rays near the sun. Nonetheless, his arrival in the German capital received enough attention that the popular newspaper Vossische Zeitung commissioned him to write a brief article to present his theories to the public.

C. Russo Krauss () University of Naples Federico II, Naples, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Russo Krauss, L. Laino (eds.), Philosophers and Einstein’s Relativity, Boston Studies in the Philosophy and History of Science 342, https://doi.org/10.1007/978-3-031-36498-3_2

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In this short article, after introducing the main points of his theories, Einstein declares that «it is impossible to enter here into an in-depth discussion of the erkenntnistheoretischen and naturphilosophischen assumptions and consequences» of the relativity of space and time; however, for «those who want to familiarize themselves with a more detailed substantiation and justification», he recommends reading either the paper of the physicist Emil Cohn titled Physikalisches über Raum und Zeit, or Joseph Petzoldt’s Die Relativitätstheorie der Physik (Einstein [1914] 1997, 4). Despite an unsuccessful academic career, in which he only obtained a position as Privatdozent at Berlin Technische Universität, at the time Joseph Petzoldt had a well-established position in the German philosophical milieu. He had already published several works, he was a vocal follower of Ernst Mach and Richard Avenarius in the philosophical circles of the capital, and he was also the founder of the recently established Berlin Gesellschaft für positivistische Philosophie, whose founding manifesto had been signed by such figures as the psychiatrist Sigmund Freud; the mathematicians Georg Helm, David Hilbert and Felix Klein; the positivist historian Karl Lamprecht; the evolutionary biologist Wilhelm Roux; the botanist Henry Potonié; the philosophers Wilhelm Schuppe, Theodor Ziehen and Wilhelm Jerusalem; the sociologist Ferdinand Tönnies; the experimental psychologist Georg Elias Müller; the physiologist Max Verworn; as well as Einstein himself (cf. Hentschel 1990a). Petzoldt’s interest in the philosophy of Mach and Avenarius dated back to his university years. After reading their works, he started a correspondence with Avenarius, that soon developed into a friendship that lasted until Avenarius’ death in 1896. Petzoldt also exchanged some letters with Mach, but the personal relationship with him grew closer only after the turn of the century. In 1898 Mach suffered a stroke that left him hemiplegic and burdened with serious health issues. For this reason, Mach decided to retire in 1901 and tried – without success – to get Petzoldt appointed as his successor for the chair of “Philosophy of the inductive sciences” in Vienna. Believing that he did not have long to live, Mach also entrusted Petzoldt with the task of editing all future editions of his most successful book, The Science of Mechanics. In the first decades of the century, the correspondence between Mach and Petzoldt intensified, and the latter traveled several times to visit his mentor. Unable to participate directly in the scientific and philosophical community, Mach relied on Petzoldt and a circle of friends to keep in touch with the cultural environment (cf. Wolters 1987, 174–75). In summary, in 1914 Petzoldt was a recognized representative of Ernst Mach’s thought. Moreover, he was the second philosopher to deal with the theory of relativity (after the neo-Kantian Paul Natorp), and his work on the new physical theory also received an endorsement by Einstein himself. It is no surprise, then, that in the following years Petzoldt carved out an important role for himself in the ever-growing debate on the philosophical implications of Einstein’s conceptions, by acting as the main supporter of the Machian interpretation of the theory of relativity. All the more so, considering that Mach never published anything on Einstein’s theories, thus leaving the field open for his pupil.

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The role played by Petzoldt in the debate about relativity has been addressed before. The thorough philological investigation by Gereon Wolters reconstructed the relationship between Petzoldt and Mach in order to disprove the false idea that Mach rejected Einstein’s theory (Wolters 1987). Don Howard highlighted the influence of Petzoldt’s notion of Eindeutigkeit on Einstein’s own epistemological conceptions (Howard 1992). And Klaus Hentschel included Petzoldt’s position in his broad examination of the discussion on the philosophical interpretations (and misinterpretations) of relativity (Hentschel 1990b). However, all these studies dealt with Petzoldt starting from different concerns. Consequently, they focused on his thought only to the extent necessary to achieve their primary goals. Conversely, the purpose of this paper is to address directly the figure of Petzoldt, following the intellectual journey that led him to his interpretation of the theory of relativity. The goal of this analysis is to show that, although Petzoldt was a follower of Mach, and – in a certain sense – precisely because he wanted to establish a rigid philosophical system on Machian grounds (which is in contradiction with Mach’s basic philosophical attitude) – Petzoldt’s and Mach’s thoughts have some fundamental differences. For this reason, their connections to Einstein’s work overlap only partly. Adopting the useful terminology proposed by Klaus Hentschel in the paper presented in this book, we may say that it is problematic whether Petzoldt and Mach belonged to the same “interpretational frame”. Even though they clearly belonged to the same Denkkollectiv (thought collective, Fleck [1935] 2012) and thus shared some assumptions and conclusions – both with respect to general epistemology and to the specific topic of the interpretation of Einstein’s theories – there is a “core assumption” that separates them: Petzoldt believed in the complete determination of all natural phenomena, which he tried to express through the notion of Eindeutigkeit (univocalness), whereas Mach always maintained a certain degree of indeterminism. In order to show the specific features of Petzoldt’s thought, we will first present his philosophical system, which he named “relativistic positivism”. Afterward, we will go into his discussion with Mach about the relativity of space and time, which took place before the two learned of Einstein’s theories. Finally, we will examine his interpretation of the theory of relativity and his checkered relationship with Einstein.

2.2 Joseph Petzoldt’s Relativistic Positivism When Petzoldt discovered the works of Mach and Petzoldt between 1883 and 1884, he was looking for a philosophy capable of overcoming the rampant “Kantian agnosticism”, i.e. the idea that we are stuck with our representations and unable to ever fathom what lies beyond them (see for example F. A. Lange [1865] 2010; Du Bois-Reymond [1872] 1874; on the topic see also Köhnke 1991; Beiser 2017). This conception not only undermined our hope to truly know the world, but also implied a dualistic split between reality on the one hand and the mental domain on the other hand.

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In contrast to mainstream philosophy, which starts from the opposition between subject and object, outer and inner world, Avenarius and Mach based their thought on the unity of experience. According to them, there is no such thing as a fundamental opposition between the physical and the psychical. On the contrary, experience is fundamentally uniform. Since empirical contents have countless relations and reciprocal interactions, the difference between psychical and physical contents simply boils down to the different contexts and relations in which one and the same content may occur. As Mach famously said: That traditional gulf between physical and psychological research, accordingly, exists only for the habitual stereotyped method of observation. A color is a physical object so long as we consider its dependence upon its luminous source, upon other colors, upon heat, upon space, and so forth. Regarding, however, its dependence upon the retina [ . . . ], it becomes a psychological object, a sensation. Not the subject, but the direction of our investigation, is different in the two domains (Mach [1886] 1897, 14–15).

Mach and Avenarius not only refuted the split between physical and psychical contents, but they also believed that mental activity is not governed by a peculiar set of laws. For them, the epistemological principles that determine our knowledge are not transcendental, or worst yet metaphysical, but have their roots in the biological organism, with its physiological needs and its Darwinian evolution. For this reason, there is no leap between the different kinds of knowledge; on the contrary, a continuous thread links the simple perceptions of animals, the first conceptualizations of early humans, and the highest abstract thinking of modern scientists. Petzoldt followed in the footsteps of Mach and Avenarius, and tried to further develop their conceptions into a complete philosophical system. First of all, he strengthened the unitary tendency of their thought even more. According to him, mental activity is not only rooted in the biological functioning of the organism, but it is subject to the same laws of all reality. For Petzoldt, the same principles apply to the inanimate world, to living beings, and to the mental domain. The principle of tendency towards stability is such an all-encompassing principle. Petzoldt adopted this principle from Gustav Theodor Fechner, who stated that due to the effects of internal forces, in the long run all systems tend toward more stable states (Fechner 1873; see also Heidelberger 2004, 8). Similarly, Petzoldt’s stability principle affirms that all systems tend to maintain or restore their equilibrium, by eventually reducing the disturbing elements to a minimum. For example, when an outer celestial body enters the solar system, it produces some disturbances in the regular orbits of the planets; but, eventually, the celestial body and the planets will develop a new regular course. Analogously, organisms constantly maintain their equilibrium by processing stimuli and nutrients coming from outside. Likewise, our mental activity consists in the constant elaboration of new and/or unsettling data through more familiar concepts, and in the development of more usual concepts to elaborate such data (Petzoldt 1887, 1890). Petzoldt designed the principle of tendency towards stability to subsume Avenarius’ principle of the least amount of energy and Mach’s principle of thoughteconomy under a higher principle. However, he also believed that his principle

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contributed to eliminate the teleological remnants left in the conceptions of Mach and Avenarius. According to Petzoldt, the whole idea of “saving” energy was nonsensical. Since every process that occurs in the world uses precisely the energy that is within its power, there can be neither an over-expenditure nor an underexpenditure of force. Therefore, Avenarius, and Mach’s economical conception of knowledge revealed the underlying idea of a finalistic drive towards optimization. Conversely, for Petzoldt the stability principle was based on the simple (and almost self-evident) fact that unchanging systems do not change, and changing systems do change, until they change into systems that do not change anymore; so that all systems eventually change into systems that do not change, i.e. stationary systems. For Petzoldt, not only there is nothing finalistic in the principle of tendency towards stability, but there is nothing finalistic in the principles that describe our mental activity either. For this reason, he criticized Mach and Avenarius for (occasionally or unwillingly) slipping back into the description of knowledge as a struggle in which a limited mind strives to grasp the world, for this conception implies a mind capable of setting its own goals. On the contrary, Petzoldt conceived knowledge as a natural process; more precisely, as a relation between various elements of the world: ourselves, (i.e., our bodies and brains) and the objects of the environment. Consequently, for Petzoldt knowledge does precisely what is within its power, according to the laws that govern it, the brain, and the rest of the universe. The last major improvement that Petzoldt wanted to bring to Mach and Avenarius’ thought regarded precisely the just mentioned lawfulness of natural processes. Since both thinkers remained faithful to their Humean legacy, they claimed that we can only experience a certain constancy in natural phenomena, but we do not experience any necessity in them. For this reason, Petzoldt believed that they left room for a certain degree of indeterminism. Therefore, even though he fully embraced Mach’s and Avenarius’ notion of “functional relations” as a mean to overcome the old metaphysical-anthropomorphic idea of the cause acting on the effect (see Heidelberger 2010), he also wanted to recover the concept of natural necessity. For this purpose, he developed the principle of Eindeutigkeit, which can be translated as “uniqueness” or “univocalness”. According to Petzoldt, we must replace the outdated notion of natural necessity, according to which the cause compels the effect to occur, with a new notion, that preserves only the empirical and scientific content of natural necessity: i.e. the fact that, given a set of conditions, only one outcome will occur, among the infinite possible ones. For Petzoldt the real case is unique, and it is univocally determined by the pre-existing conditions, which means that these conditions lead to only one outcome. Consequently, the aim of science must be to describe natural phenomena in such a way that addresses the real case in its uniqueness (Petzoldt 1890, 215–16; 1895). In his own words: For every process means of determination [Bestimmungsmittel] may be found that determine it uniquely [eindeutig], in such a manner that for every variation in this process that one can conceive as being determined by the same means, one can find at least one other variation that would then be determined in the same way as, and thereby be equivalent to, the first, and thus would have, as it were, the same right to be realized as the first (Petzoldt 1900–1904, 1:39; for the translation see Howard 1992, 170; see also Petzoldt 1895, 168).

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To put it differently, the scientific “means of determinations” provide a single (univocal, eindeutig) model for the real case, and multiple equivalent (ambiguous, equivocal, or, more precisely, plurivocal, vieldeutig) models for the merely possible cases. For example, when we use the law of the parallelogram of forces to describe the motion of a body, we use some means of determinations (the vectors) to describe the real case in its uniqueness. Since the diagonal of the parallelogram is unique, unlike all the other infinite possible lines that depart from the point of impact, it can be used to describe the uniqueness of the real motion of the body (cf. Petzoldt 1900–1904, 1:34 ff.). Like the stability principle, the principle of Eindeutigkeit too applies to the whole of reality, including mental activity. However, for Petzoldt there is no psychical causality, which means that mental contents do not depend on each other, they are not univocally determined by other mental contents. Hence, Petzoldt believes that the only way to regard psychical activity in terms of univocal determination is by connecting it to brain activity. In other words, the cerebral processes are the conditions and “means of determination” to explain why a certain psychical event necessarily had to occur at a given moment, instead of another of the countless possible ones. Therefore, like Mach and Avenarius, Petzoldt too believed that the physiological activity of the brain is the root of our cognitive functions. In particular, he even claimed that it provides us with a strong foundation for the principle of Eindeutigkeit. Indeed, the brain is a biological organ that is the result of a long evolution guided by the principle of stability: the nervous systems (or sub-systems) that develop a stable relationship with the environment survive, whereas the others either continue to change or die out. Consequently, the development of the brain is shaped and determined by the environment. This means that the reason why we feel a demand for Eindeutigkeit is because Eindeutigkeit is embedded in the very physical constitution of the brain, since Eindeutigkeit is a feature of the natural environment in which the brain evolved. To put it schematically: (1) nature is univocally determined; (2) this characteristic of nature enables the evolution of the brain (or, more precisely, evolution in general, since an ever-changing chaotic nature would prevent the development of any kind of stable organic system); (3) the brain evolves in accordance with a univocally determined nature; (4) hence, in order to survive, the brain needs nature to stay univocally determined; (5) therefore, the brain demands nature to be univocally determined; (6) since mental activity depends on cerebral activity, mental activity too needs univocal determination and hence posits univocal determination as a postulate; (7) finally, since nature is univocally determined in the first place, we see our demand for univocal determination fulfilled by nature. And thus the circle is complete. With this conception, Petzoldt believed to have overcome the Humean skepticism of his masters, by providing a firm empirical basis for the belief in natural necessity: The strength of our principle [of Eindeutigkeit] does not come from a sum of single experiences, but from the fact that we demand its validity from nature. Before it is a law, it is a principle by which we approach reality, a postulate. It is comparatively a-priori

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valid, independent of any single experience. But it would be bad for a philosophy of pure experience to teach about a-priori truth and to fall back in the most sterile metaphysics. [ . . . ] The strength of our principle has its roots in very general experiences [ . . . ]. These are the facts of our own existence and of the existence of the world; the fact that we are thinking and acting beings; the fact that there is evolution. None of this would be possible without the complete determination of natural events, which is the most general necessary condition for all this (Petzoldt 1900–1904, 1:40).

Let us sum up the main points of Petzoldt’s thought that we have discussed so far, before introducing how he shaped them into his philosophical system: relativistic positivism. (1) He agrees with Mach and Avenarius that there is no duality between the physical and the psychical, since empirical reality is fundamentally homogeneous and unitary. (2) He affirms the existence of an all-encompassing principle of tendency towards stability according to which all systems (be they inorganic, organic, or mental) eventually become stable systems; (3) He develops the principle of Eindeutigkeit, which claims that all events in the universe (again, be they inorganic, organic, or mental) are univocally determined, which means that there must be some set of conditions, and therefore some means of determination, which account for the fact that only this one event had to occur rather than one of the other countless possible ones. Now, according to Petzoldt, “positivism” is the philosophy that rejects the notion of “substance”, whereas “relativism” is the philosophy that refutes the opposition between reality and appearances. This means that the fundamental idea of relativistic positivism is that there is no underlying unchanging substratum that is supposed to be “the real thing” beyond all the changes and relations that are supposed to be the “appearances” of the thing. Rather, everything is what it is in its relations to other things. Consequently, things are variable and many-sided: an object may be big and small, green and grey, hot and cold, because it always has these (seemingly contradictory) properties in relation to other objects. In particular, according to relativistic positivism, our experience is an example of the relations that exist in the world. In contrast to traditional philosophy, which regards our experience as a mental image (a representation, an appearance of the world), Petzoldt stresses that the world and our experience of it are not two separate realms. We are a part of the world, and our experience is the relation between ourselves (our body, more precisely our nervous system) and the other objects of the world. Petzoldt points out that relativistic positivism should not be equivocated for some sort of phenomenalism, or skepticism, or even solipsism, according to which we are stuck with our limited point of view, with our subjective perspective, and therefore cannot know the world. Since there is no thing-in-itself, no substance, but only relations, knowledge of the world in relation to us is knowledge of the world. In short: real knowledge is relative knowledge. When we state that a thing has a certain property in relation to us, we are stating a fact about reality, not just about an appearance. Even more importantly, we should not forget that relations are univocally determined, and univocal determination provides us with the sort of stable knowledge

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that we need. As we have seen, it is perfectly natural for us to want the world to be stable, but for Petzoldt we should not satisfy the need for stability by looking for an immutable substance that underlies all change and all relations. Rather we should look for the stable necessity that governs all change and all relations. Univocal determination not only gives us stability, a firm understanding of the world. By means of univocal determination, we can also grasp a portion of the network of innumerable necessary relations that exist in the world; in this way, we can establish connections between different perspectives, so as to bridge the gap that would otherwise separate different points of view. To illustrate this point, let us consider the example of the human experience. As we have already said, human experience is the relation between our bodies and the objects of the world, and this relation is univocally determined. In particular, it is univocally determined by the physiological processes that take place in the body and in the nervous system. So, when I say “I see a green tree”, it actually means that there is a functional relation between the green tree and my eyes, or – more generally – my perceptual apparatus. Let us imagine a colorblind person, who might say “I see a grey tree”. This means that there is a functional relation between the grey tree and the eyes or perceptual apparatus of the colorblind person. What matters is that, for Petzoldt, we do not need a tree-in-itself to regard these two experiences as referring to the same object; simply, the grey tree and the green tree are the same tree in relation to different individuals. According to traditional philosophy, there is a fundamental gap between my experience of the tree and the other person’s experience of the tree, as well as between our experiences of the tree and the treein-itself. Conversely, Petzoldt believes that there is no tree-in-itself, and that the tree in relation to me is green, and the tree in relation to the colorblind person is grey, precisely because there is no tree-in-itself, but only the tree as it is in innumerable relations to all other objects of the world, including my body and the body of the colorblind person. However, the most important thing is that once we know the functioning of human bodies (or, more precisely, once we know the univocal determination that governs the functional relations between human bodies and the objects of the world) we know that a person with a certain bodily constitution must necessarily perceive the tree as green, and a person with a different bodily constitution must necessarily perceive the tree as grey. Consequently, we know as a matter of scientific fact what each person must experience; we know how the world must be from their perspectives, in relation to them. Hence, there is no unbridgeable gap between the different sensory experiences of different people, because univocal determination allows us to know scientifically how the world is in relation to different bodies, in the eyes of different people. That is why Petzoldt, in the wake of Mach, believed that psychophysics could play a fundamental role in the forthcoming development of science. Psychophysics had the task of bridging the gap between the different fields of science. Instead of having physics as the science of the supposed real material world on the one hand, and psychology as the science of the inner world of appearances on the other hand, we would have only one unified science, investigating the innumerable

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relations that exist in the empirical world, looking for univocal determination in all natural phenomena (inorganic, organic, and mental). And such an investigation would include the efforts of psychophysics to understand the necessary relations between our nervous system and the objects of the world. To recap Petzoldt’s position, let us quote this passage, which also leads us to the next section of our paper, which is about Petzoldt’s and Mach’s debate on the relativity of the frame of reference in physics: Never was a more profound truth spoken than that of Protagoras: the world is for each person as it appears to him. [ . . . ] This does not erase the lawful connection of things, nor the firm order of human affairs. We conceive the diversity of the worldviews as lawful insofar as we regard it as dependent on the diversity of the individuals. The easiest way to understand our position is to compare it with the description of the relations of motion of celestial bodies. Just as there is no absolutely stationary origin of coordinates for such description, we have no normal-intelligence whose perceptions we can regard as absolutely correct. And just as every one of those descriptions in itself provides a completely lawful picture, regardless of whether the Earth, or the Sun, or Jupiter is taken as starting point, and just as every one of those pictures is thoroughly harmonized with all the others, so the worldview of each individual intelligence (which depends on the individual organization of each one) is in itself without contradictions and is in harmony with the worldviews of all the others (Petzoldt 1906, 144–45 emphasis mine).

From these words, it is apparent why Petzoldt regarded his relativism as a form of positivism. Thanks to the principle of Eindeutigkeit, the idea that everyone knows the world from the own perspective does not shatter reality into an infinity of contradictory worldviews. Quite the opposite. The univocal determination of phenomena preserves a knowable reality and provides a framework that embraces all individual points of view. However, this knowable reality is not the old-fashioned absolute (literally: untied from all relations) reality beyond experience. Rather, reality is the lawful univocal connection of all relations (and our experience is part of these relations).

2.3 Before Einstein. The Debate with Mach About the Frames of Reference Even though Einstein published the paper on special relativity in 1905, it took a while for Petzoldt and Mach to learn of the new theory. However, both were already involved in the ongoing debate over the flaws of Newton’s account of space, time, and inertia. According to Newton, inertia is a fundamental property of bodies, which persevere in either a state of rest, or uniform rectilinear motion (Newton [1687] 2016, def. III). This means that, even if there were only a single body in the whole universe, it would still possess an inertial force. Consequently, one should admit the existence of an absolute space, otherwise, it would be impossible to indicate in relation to what that single body would be at rest or moving. As a result, Newton distinguished between absolute motion with respect to absolute space, and relative

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motion with respect to other bodies. To illustrate the difference between the two, he presented the famous example of the rotating bucket filled with water. At first, the water is moving relative to the bucket but is absolutely at rest, as proved by the absence of centrifugal force (which is an expression of inertial force). After a while, the water is at rest relative to the bucket (as they rotate jointly), but is in absolute motion with respect to absolute space, as proved by the concave shape it assumes when the centrifugal force pushes it towards the sides of the vessels (Newton [1687] 2016, Scholium, 58–59). In his Science of Mechanics, Mach criticized Newton’s account of inertia and absolute space on the basis of two arguments. First, he stressed the impossibility of detecting absolute space, which makes it a metaphysical and practically useless concept. He wrote: “if we take our stand on the basis of facts, we shall find we have knowledge only of relative spaces and motions” (Mach [1883] 1919, 232). Second, he rejected the assumption that inertia and centrifugal force (which seem to make the concept of absolute space necessary) would actually exist without respect to other bodies. The law of gravity describes the forces (i.e. accelerations) that result from the interaction of two or more bodies. However, Mach pointed out that we do not and cannot know whether the behavior of a body in the absence of forces (i.e. inertia) also depends on its relations to other bodies. More precisely, since we always observe the universe as a whole, and never a single body in isolation, we are more justified in assuming that inertia and centrifugal forces too depend upon the presence of other bodies. Accordingly, Mach reinterpreted Newton’s thought experiment of the bucket, and stated that it only teaches us that: “when a body moves relatively to the fixed stars, centrifugal forces are produced; when it moves relatively to some different body, and not relatively to the fixed stars, no centrifugal forces are produced” (Mach [1883] 1919, 543 emphasis mine). In summary, Mach claimed that reference to other bodies, such as the fixed stars, is unavoidable both kinematically, because we need a reference body to describe the motion of an object, and dynamically, because we must assume that the behavior of a body in the presence or absence of forces depends upon its relation to other bodies. This is what he meant when he wrote: “the masses that [ . . . ] exert forces on each other, as well as those that exert none, stand with respect to acceleration in quite similar relations. We may, indeed, regard all masses as related to each other” (Mach [1883] 1919, 236 emphasis mine). Upon this basis, in the fifth edition of the book, Mach formulated his research program, concerning the determination of “principles of the whole matter, from which accelerated and inertial motion result in the same way” (Mach [1883] 1919, 296 emphasis mine; see also Wolters 1987, 49 ff.). As is well known, Einstein later adopted this idea that gravity and inertia might be “identical in nature” and both “completely determined by the masses of the bodies”, and named it “Mach’s principle” (Einstein [1918] 2002, 33–34). Mach was not the only thinker to question the foundations of Newtonian mechanics in the second half of the nineteenth century. As early as 1870, the mathematician Carl Neumann criticized Newton for introducing the notion of inertia without first defining uniform linear motion. Consequently, Neumann stressed that

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the first step in establishing mechanics should be the assumption of “a special body in the universe” that serves as the point of reference for all motion (Neumann 1870, 15). Hence, Neumann preserved the idea of absolute motion, but instead of basing it on the notion of absolute space as some sort of infinite, empty vessel, he based it on an absolute body of reference, which he named “body Alpha”. Moreover, since Neumann stressed that – following Galilean relativity – any body in uniform linear motion could play the same role as the body Alpha, he “came close, though perhaps in a clumsy way, to postulating the existence of an infinite set of dynamically equivalent inertial frames” (Disalle 1993, 349). Like Newton, Neumann too elaborated a thought experiment to justify the idea of absolute motion. Let us imagine a star made of some fluid material rotating around its axis. Because of centrifugal forces, the star has an ellipsoid shape. Now, Neumann asks, what would happen if every other celestial body in the whole universe suddenly disappeared? If we reject the notion of absolute motion, we should assume that the star would suddenly stand still, because there would be nothing in relation to which the star would move. As a consequence, we would have to conclude that the shape of the star would suddenly change from an ellipsoid to a sphere. But, since this is absurd, Neumann concludes that absolute motion must be real, and to define it we need a body Alpha (Neumann 1870, 27). Neumann discussed not only Newton’s account of space but also that of time, thus inspiring the work of another thinker, Ludwig Lange. Neumann pointed out that we always measure time intervals by using as reference the motion of some other body. Hence, we cannot tell whether a single body is moving at a constant velocity because we lack an indicator for time. Vice versa, once we have two bodies in motion, the inertial motion of the first body may serve as a temporal reference for the second. Lange praised Neumann for positing as the fundamental measure of time an “inertial timescale” defined “through the motion of a point left to itself”, and tried to apply a similar strategy to define spatial coordinates (Lange [1885] 2014, 253). In particular, Lange stated that “in exactly the same way as the one-dimensional inertial timescale could be defined through one single point left to itself, the threedimensional inertial system can be defined through three points left to themselves” (Lange [1885] 2014, 253). In particular, since we need three points to define a spatial frame of reference, we can say something about the physical behavior of a single point only if we have four or more points. In other words, since for three points it is always possible to draw such a reference system that they are said to move inertially, the inertial motion is “pure convention for three such points, but embodies a noteworthy research result only insofar as it is valid for more than three” (Lange [1885] 2014, 253). In the subsequent editions of his Science of Mechanics Mach added some references to the work of Neumann and Lange. He praised Neumann’s hypothesis of the rotating fluid star as the “the most captivating reasons for the assumption of absolute motion”, but still dismissed it as a “too free a use of intellectual experiment”, since it did not involve the modification of “unimportant circumstances”, but rather the bold assumption that “the universe is without influence on the phenomenon here in question” (Mach [1901] 1919, 572). Mach spoke even more favorably of Lange,

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whose treatise he regarded as “one of the best that have been written” on the topic of frames of reference, thanks to Lange’s “careful analysis and study, from historical and critical points of view, of the concept of motion” (Mach [1901] 1919, 544). Moreover, Mach confessed that Lange’s attempt to mathematically define and construct a system of space and time coordinates “especially appeals to my mind, as a number of years ago I was engaged with similar attempts”; however Mach also added: “I abandoned these attempts, because I was convinced that we only apparently evade by such expressions references to the fixed stars and the angular rotation of the earth” (Mach [1901] 1919, 546). As we have seen, Mach believed that the fixed stars are not only our implicit frame of reference for the kinematic description of motion, but are also necessary for a dynamical account of motion. Consequently, he claimed that “it is quite questionable, whether a fourth material point, left to itself, would, with respect to Lange’s ‘inertial system’, uniformly describe a straight line, if the fixed stars were absent” (Mach [1901] 1919, 546). Petzoldt entered the debate in 1895, by trying to address two issues: absolute space; and whether the law of inertia is empirical or a-priori. On the latter, Petzoldt stressed that the law of inertia contains different statements, with different epistemological statutes. First of all, as long as it affirms that a body cannot leave its state of motion or rest by itself, it is a consequence of the principle of Eindeutigkeit, since it is just another way of asserting that every change of state must be univocally determined by some conditions. Second, it is a conditional proposition stating that if a body moves in uniform linear motion, then it will continue to move in the same way unless some external condition alters its state. Finally, oddly enough, according to Petzoldt the law of inertia also affirms that, after “a single push”, a body moves with uniform linear motion (Petzoldt 1895, 189). This claim is quite odd because the law of inertia describes what happens to a body on which no forces act, so that the case of a “push” seems incongruous. However, Petzoldt believed that in such a case the rectilinear motion is univocally determined since the straight line is unique, being the shortest connection between two points. On the other hand, for Petzoldt the uniform motion must be empirically ascertained, since a uniformly accelerated motion would be just as univocally determined as the uniform motion. Hence, the result of Petzoldt’s reasoning was that the law of inertia is partly a postulate with the same firm foundation as the principle of Eindeutigkeit, partly a conditional proposition, and partly an empirical proposition. In the third edition of the Science of Mechanics, Mach mentioned Petzoldt’s position about an “empirical and a supra-empirical element in the law of inertia”, but he also distanced himself from his pupil: I believe I am not at variance with Petzoldt in formulating the issue here at stake as follows: It first devolves on experience to inform us what particular dependence of phenomena on one another actually exists, what the thing to be determined is, and experience alone can instruct us on this point. If we are convinced that we have been sufficiently instructed in this regard, then when adequate data are at hand we regard it as unnecessary to keep on waiting for further experiences; the phenomenon is determined for us, and since this alone is determination, it is univocally determined (Mach [1901] 1919, 562).

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Even though Mach’s words downplay the disagreement with Petzoldt, it is apparent that they defended two different positions: according to Mach, the univocal determination is simply the result of inductive generalization, whereas for Petzoldt it is a fundamental postulate engraved into us by evolution. For this reason, Petzoldt himself stressed that he disagreed with Mach’s assertion that “all forms of the law of causality spring from subjective impulses, which nature is by no means compelled to satisfy” (Mach [1883] 1919, 586 emphasis mine). Indeed, Petzoldt believed that “our own existence is proof that there definitely is such a necessity” (Petzoldt 1895, 486). Concerning the issue of absolute space, Petzoldt shared “Mach’s and Neumann’s objections” to Newtonian mechanics, but he also claimed that we should “regard ourselves as the reference body” (Petzoldt 1895, 192 emphasis mine). Or, more precisely, he affirmed that we already implicitly use ourselves as the frame of reference, since, even “when we try to mentally remove from space all other bodies, except the one whose motion we are considering, this can never happen for our own body” (Petzoldt 1895, 192). Accordingly, Petzoldt re-interpreted Newton’s bucket argument by stating that all motions involved in the thought experiment are relative, simply – contrary to what Mach said – not relative to the fixed stars, but relative to us. Therefore, we do not need the fixed stars to regard the rotation of the bucket and the water as univocally determined (Petzoldt 1895, 193 n.). The fact that Petzoldt believed he could substitute Mach’s fixed stars with our own bodies indicates that he failed to understand that Mach was not only talking about the “relativity of representation”, i.e. the standpoint from which a certain phenomenon is observed and described. Rather, Mach was also talking about the “relativity of bodies”, i.e., the fact that physical processes are determined by the mutual relations between the masses of the universe (cf. Wiechert 1992, 166). Even if one accepts that one’s own body can be used for a kinematic description of motion, it is much more difficult to sustain that it may play the same role as the fixed stars in a dynamic description of motion, since they have wildly different masses. Hence, it is no surprise that in his reply to Petzoldt Mach stated: I cannot understand how all the physical difficulties involved in the present problem can be avoided by referring motions to one’s own body. On the contrary, in considering physical dependencies abstraction must be made from one’s own body, so far as it exercises any influence (Mach [1883] 1919, 571–72 emphasis mine).

The discussion between Mach and Petzoldt continued in their private correspondence. After the publication of the 1901 edition of the Science of Mechanics, in which Mach addressed Petzoldt’s position, the latter wrote a long letter to his mentor, in which he took stock of their agreements and disagreements. In the letter, Petzoldt “concedes” that “physically speaking” one can only “relate the motion of a body to the Earth or the fixed stars”; but, at the same time, he does not agree that “with the physical question all the question about the law of inertia would also be settled”, for “erkenntnistheoretischen” questions remain (Petzoldt 1901 emphasis mine). Indeed, Petzoldt claims that “problems like Newton’s bucket or Neumann’s celestial body are no longer physical problems”; but for this very reason they may help us to clarify the just mentioned erkenntnistheoretischen questions. Therefore,

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we should not hastily “draw the line between the allowed and the no-longer-allowed use of the thought experiment”, at least as long as one stays true to “the laws of thought and the demand for Eindeutigkeit” (Petzoldt 1901). To prove his point, in the letter Petzoldt takes up Neumann’s thought experiment, adjusting it for his own purposes. Again, we have a fluid star rotating around its axis and shaped like an ellipsoid due to centrifugal forces. Now, Petzoldt imagines that we remove all the celestial bodies of the universe, one by one. The fluid star would still be an ellipsoid rotating relative to the remaining celestial bodies. Eventually, only the rotating fluid star and another celestial body, however small, would remain. What would happen if we removed this last – no matter how tiny – celestial body? Apparently, the fluid star would suddenly become a sphere, since it would stop rotating relative to something, and we cannot assume absolute motion. However, for Petzoldt this idea is “intolerable” because there is a conflict between the suddenness of the change of shape and the continuity of the process of elimination of the other celestial bodies. Nevertheless, unlike Neumann, Petzoldt does not rely on this example to argue in favor of absolute motion. Similar to Mach, he believes that the “basis of this paradox” is that “abstraction has gone too far”; although, “not the abstraction from the fixed stars, but from our own body” (Petzoldt 1901). We could dismiss Petzoldt’s argument and emphasize that he – once again – overlooks the fact that Mach regards the fixed stars as indispensable not just as a kinematic reference, but also because – in the absence of any (impossible) evidence to the contrary – we must assume that their masses are a condition of inertial motion. So, we cannot assume that the progressive elimination of all celestial bodies, and the subsequent change in the masses of the universe, would not affect the centrifugal forces in the rotating fluid star up until the moment when we reach the elimination of the last remaining tiny celestial body. Indeed, in his discussion of Newton’s bucket argument, Mach presented a specular argument, stating that “No one is competent to say how the experiment would turn out if the sides of the vessel increased in thickness and mass till they were ultimately several leagues thick” (Mach [1883] 1919, 232). Nevertheless, there is still a more charitable reading for Petzoldt’s reasoning. What Petzoldt is trying to say is that the revised version of Neumann’s thought experiment helps us envision the contiguity and difference between physical and epistemological issues. The question “What would happen to the rotating fluid star if we removed the other celestial bodies from the universe?” may be regarded as a physical question. However, if we continue along this path we eventually reach the question of the elimination of the last remaining body, which is indeed an epistemological issue, that involves the conceivability of something absolute. According to Petzoldt, we cannot really think a single celestial body in absolute isolation. When we try to do that, we simply forget ourselves and the fact that it takes a nervous system to register or imagine the presence of that celestial body. Therefore, Petzoldt writes: Sure enough, I should not prescind with impunity from it [my body] when it comes to ultimate questions. And here we are dealing with such an ultimate question, with the epistemological exclusion of mechanics from the rest of our knowledge; with the not quite insignificant question of whether our abstraction can continue even when we, through a

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continuous path, cross beyond the narrower field of physics, or whether there is a point of discontinuity here, which we must enclose in a fence; [the question] whether the laws of rotational motion – that we must regard as independent of any part of the fixed stars and of any combination of these parts, and upon which we instinctively do not allow the influence of any celestial body – can be maintained down to the very last conceivability. Is it not natural that the special sciences come across questions which compel us to cross beyond or remove our arbitrarily drawn borders? (Petzoldt 1901).

Indeed, in the revised version of Neumann’s thought experiment, the gradual elimination of the celestial bodies shifts the question from the physical problem of the interaction between bodies to the epistemological problem of the conceivability of bodies in themselves. The thought experiment takes physics to its limits, to the question whether this science can represent the world per se, independent of any observer. A positive answer would mean, as we just read, the “exclusion of mechanics from the rest of our knowledge”, which – as we know – is always about relations. Therefore, if Mach accepts the idea that we can leave ourselves and our bodies out of consideration, he ends up validating precisely that belief in the exceptionalism of physics that he has fought against all his life. Moreover, Mach himself always insisted that physics should keep in touch with the concrete empirical data upon which it is based. Hence, he should agree that there is no empirical data without us, since our own body is the key variable in all our relations to the objects of the world. In Petzoldt’s words: “We move away from original experience even less if we abstract from the totality of celestial bodies than when we abstract from our own body” (Petzoldt 1901). Besides, as we have seen, Mach stated that “in considering physical dependencies abstraction must be made from one’s own body, so far as it exercises any influence”. Hence, Petzoldt responds as follows: With the same right (or even more right) with which you claim “but it is not to be antecedently assumed that the universe is without influence on the phenomenon here in question” (Mach [1883] 1919, 596), one may assert: “that the I is without influence . . . ” (Petzoldt 1901).

Moreover, Petzoldt also emphasizes that Mach himself had stated in his Analysis of Sensations that “A magnet in our neighborhood disturbs the particles of iron near it; a falling boulder shakes the earth; but the severing of a nerve sets in motion the whole system of elements”, so that we may say that our body “has a more extensive and profound action” on the world than the rest of the objects (Mach [1886] 1897, 14). Therefore, we cannot simply assume that our own body is “without influence”, because “this is often not the case, not even in physical investigations” (Petzoldt 1901). In fact, “for mechanics the body as a whole as well as in its parts is no less a physical object than any other celestial body, hence it can easily take on the role of that last fixed star” (Petzoldt 1901). Let us try to sum up and organize Petzoldt’s arguments. (1) It is true that physics focuses mainly on the relations between bodies, leaving out any reference to our own body. Nonetheless, we cannot assume from the outset that our own body has no physical influence. Rather, such a statement should at best be the result of an empirical investigation. (2) Kinematically, we always need a reference body to

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describe motion. In this case, our own body has the advantage over fixed stars that we always visualize the spatial relations from our actual observation point, i.e. our body. (3) From an epistemological perspective, we can never overlook our own body, not even in the case of physics, because our knowledge is univocally determined by the cerebral substrate and its relations to the world. Hence, to claim that we can take our own body out of the equation would mean to fall back into the old belief in absolute knowledge. In the above discussion, we find once again two fundamental topics of Petzoldt’s philosophy: relativism and Eindeutigkeit. According to relativism, all components of reality are in mutual relations, and – since we ourselves are part of this reality – we can only know the world by observing it from our concrete point of view, i.e. from the relation between our own body (perceptual apparatus and nervous system) and the objects of the world. According to the principle of Eindeutigkeit, the goal of science is not to know the material world that lies beyond our experience, the absolute reality in itself; but to study the relations between the empirical phenomena in their univocal determination, to regard the universe sub speciae univocitatis, so to speak. Consequently, our own body can and should be used as the point of reference in physical investigations, because univocal determination prevents any risk of falling into a skeptic form of relativism. Unfortunately, we do not have Mach’s answer to Petzoldt’s letter. Moreover, in the further course of the correspondence the subject drifted towards other topics. In the following years, every now and then, Petzoldt mentions to Mach that he regrets his little training in mathematics and physics, and that he is studying to fill up these gaps. The result of this study was a paper that Petzoldt published in 1908, under the title Die Gebiete der absoluten und der relativen Bewegung (The fields of absolute and relative motion). In this work, Petzoldt distinguishes between relativism in the broader sense (i.e. philosophical relativism, according to which all knowledge is relative) and relativism in the narrower sense (i.e. the discussion in physics about the notion of absolute motion). Even though in 1908 Einstein had already published the paper on special relativity, Petzoldt’s essay does not mention this new theory, but only addresses the ideas of Mach, Neumann, and Lange. Following Mach, who in the recently published Knowledge and Error had made a distinction between “physiological space” and “metrical space” (Mach [1905] 1976, 251 ff.), Petzoldt now argues in favor of the notion of absolute metrical space. At least from a logical perspective, the concept of absolute space is entirely legitimate, since we build this notion by abstracting from the concrete visual spaces of the individuals. Hence, the question whether we can envision absolute motion is entirely different from the question whether science can and should make use of the concept of absolute motion (Petzoldt 1908, 34 ff.). For Petzoldt, the concept of absolute motion is not only logically legitimate, but also physically meaningful. Indeed, there are two conflicting hypotheses. According to Newton and Neumann, motion has two components: an absolute component, that belongs to the body itself, i.e. inertia; and a relative component, that depends on the interaction with other bodies, i.e. gravitation. According to Mach, all motion is

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relative, and inertia too depends on the relation to other bodies. However, Petzoldt points out that we do not know how a single body in complete isolation would behave: whether it would display inertial motion or centrifugal forces (like in the example of Neumann’s rotating fluid star). Therefore, for Petzoldt, we cannot decide who is right and who is wrong. Newton and Neumann, or Mach. Consequently, at least at present, absolute motion is a legitimate physical hypothesis (Petzoldt 1908, 37 ff.). Subsequently, Petzoldt engages with Lange, whose proposal he brands as simply “wrong”. As we have seen, Lange claimed that the trajectory we ascribe to the first three points is entirely conventional, and that we can only predicate inertial motion when we speak of a fourth point. However, according to Petzoldt, Lange surreptitiously restrict the movements of the three initial points: they move continuously, without jumps; they do not go back, or oscillate; they move mutually (not independently, as claimed), so that they either endlessly move away from each other, or get closer to each other until they meet, and then begin to drift endlessly apart. In other words, Lange already tacitly assumes that the three points left to themselves are moving inertially. This means that Lange, in spite of himself, agrees with Newton in regarding a single body left to itself as moving inertially (Petzoldt 1908, 45 ff.). Hence, in the chicken and egg controversy between Newton (who claims that absolute inertial motion comes first, because, once we posit a single body, we must assume that it moves inertially) and Lange (who claims that we cannot speak about inertia without first positing some other bodies in relation to which the body is moving), Petzoldt surprisingly takes sides with Newton, indeed, Petzoldt admits that it is “rather uncomfortable to posit as the foundation for a theory something that can never be experienced”, such as absolute motion. Nonetheless, “this discomfort must disappear once we remember that even the metrical space of natural sciences” is a notion that “goes beyond the limits of what is experienceable”. Consequently, both metrical space and absolute motion should be regarded as “concepts that must be justified by their performances in the description of natural phenomena” (Petzoldt 1908, 55). After arguing that absolute motion is a legitimate concept from the point of view of logic and physics, Petzoldt asks whether this notion is also “appropriate” (zweckmässig). According to him, we should always favor the scientific description that stays more true to the empirical facts, since we cannot achieve stability until there is a complete adaptation of ideas to each other, but also of ideas to facts (reference to Mach [1886] 1897, 156). Hence, we should expect physics to evolve in such a way that the abstract metrical concept of space comes closer and closer to the actual visual space of experience. From this perspective, the relative concept of motion is superior to the absolute concept of motion, because it stays more true to the experience. For the same reason, Mach’s idea that our spatial and temporal frames of reference are, respectively, the fixed stars and the rotational angle of the Earth must be regarded as superior to the positions defended by Newton, Neumann, and Lange. (Petzoldt 1908, 57).

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In this praise of Mach’s position, for the first time Petzoldt seems to understand that the fixed stars play a role not only in the kinematic description of motion, but also for a dynamical account of it. In his paper, Petzoldt now writes: The law of inertia already contains the reference to fixed stars. Indeed, the inertial motion with constant direction and velocity is the logical correlate of acceleration. If the concept of mass serves for the description of the processes of motion, then we cannot omit the fixed stars as system of reference. Hence, the concept of mass is the specific and central concept of all mechanics (Petzoldt 1908, 58 emphasis mine).

In so doing, Petzoldt finally acknowledges that the core of the issue is not (only) the need for a body of reference, but rather the dynamic role of the masses of the universe, that must be regarded as a condition both for gravitational motion and for inertial motion (Mach’s principle). From the above, it is evident that this paper is an anomalous work in Petzoldt’s intellectual evolution, since he now speaks favorably of Newton’s absolute motion. However, we can explain this apparent incongruity by regarding it as a consequence of the tension between the two fundamental pillars of Petzoldt’s philosophical system: relativism and Eindeutigkeit. Petzoldt moves closer to Newton because he cannot accept the idea that the motion of a single body might not be univocally determined, that a single point might move arbitrarily, so much so that its trajectory could be regarded as purely conventional (as Lange stated). For this reason, Petzoldt affirms that what is “fatal” to Lange’s theory is its “indeterminacy” and “lawlessness” (Petzoldt 1908, 55). For Petzoldt, relativism can never mean the absence of determinateness. Consequently, even Machian relativism is re-evaluated: the reason for referring to the fixed stars is not the fact that we cannot know how a single body would behave if there were no other celestial bodies in the universe (i.e. a negative knowledge, grounded on a lack of determinateness, on a suspension of judgment), but rather the idea that inertia is univocally determined by the masses of the other celestial bodies (i.e. a positive knowledge, although not yet entirely proven).

2.4 After Einstein. Relativity and Eindeutigkeit As anticipated, even though Petzoldt was working on the topic of absolute motion, it took him a while to notice the appearance of Einstein’s relativity. The first mention of the new theory is found in a letter he wrote to Mach in 1910: Concerning Einstein’s theory of relativity, as presented by Classen (1910), I haven’t worked enough on it yet. However, Einstein’s fundamental idea seems to be quite excellent. Although, I wonder whether he has freed himself of the absolute altogether. For example, I don’t see why the speed of light c and c’ must be equal (Petzoldt 1910; also in J. T. Blackmore and Hentschel 1985, 84–85).

The year before, Mach had published the second edition of his History and Root of the Principle of Conservation of Energy, in which he declared his agreement with Minkowski’s principle of relativity (1909), while also stressing that he had

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already expressed similar ideas in his Science of Mechanics and Theory of Heat (Mach [1909] 1911, 95). The affinity between his own position and Einstein’s and Minkowski’s theory was stressed again in the paper Mach wrote in response to Max Planck’s The Unity of the Physical World Picture (Planck [1908] 1992). According to Planck, Mach’s conceptions about the objects as bundles of sensations, and about the economic drive guiding our knowledge, were an obstacle to scientific progress. Therefore, Mach replied by pointing out that the new theories developed by Einstein and Minkowski were consistent with his own position (Mach 1910). Mach also sent the second edition of the History and Root of the Principle of Conservation of Energy directly to Einstein. We do not know what was in the accompanying letter, however, it is telling that Einstein replied to this letter by siding with Mach in the polemic with Planck: Naturally, I am well acquainted with your principal works, of which I especially admire the one on mechanics. You have had such an influence on the epistemological views of the younger generation of physicists that even your current opponents, such as, e.g., Mr. Planck, would undoubtedly have been declared to be “Machists” by the kind of physicists that prevailed a few decades ago (Einstein [1909] 1995).

As for Petzoldt, he returned on Einstein’s theories in a letter to Mach of June 1911: One step at a time, I am dealing more closely with the principle of relativity. It is always interesting how people cannot go beyond the opposition between appearance and reality, such as Berg (1910), whose exposition is otherwise excellent, and far better than those by Classen (1910) and Gruner (1910). Recently you wrote to me that you feel that something is still missing, from an epistemological perspective, in the principle of relativity. I think so too; the reason is that the authors of the principle [Einstein and Minkowski] have not yet arrived at a free view, even though in general their lack of prejudices has been demonstrated. It seems to me that the constancy of light-speed still plays a peculiar “naïve” role. Surely, it is constant in different space-time systems only thanks to a deformation, and this must be assumed from the outset. However, it seems to me that mathematics is further ahead in the foundation of geometry than theoretical physics. Maybe, instead of connecting only space and time, Minkowski could have also included the speed of light. However, maybe I don’t understand enough yet. In any case, I think that the recent development in theoretical physics and mathematics is wonderful. Thereby all absolutism and apriorism are eradicated. The old Protagoras returns from the grave and his resurrection is followed by the outpouring of the holy spirit of relativism (Petzoldt 1911; also in J. T. Blackmore and Hentschel 1985, 91).

Even though we will delve later into Petzoldt’s reading of the relativity, these words already sketch the core of Petzoldt’s interpretation of Einstein’s theory: i.e. the belief that it confirms philosophical relativism, since both lead to the overcoming of the opposition between reality and appearance. Moreover, this passage introduces an idea that Petzoldt will continue to sustain in his later writings, namely that the physicist can choose the variants and the invariants at will. In his theory, Einstein set the speed of light and the laws of nature as invariant, and therefore space and time became variant. For Petzoldt, it is possible to imagine an alternative model in which space and time play the role of the invariants, whereas the speed of light and what Einstein posited as laws of nature become the variants (cf. Petzoldt 1914, 40–41).

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Due to his growing interest in the theory of relativity, in the following years Petzoldt got closer to Einstein. First of all, he contacted him asking to join the Gesellschaft für positivistische Philosophie. Then, he started publishing a series of works dealing with the philosophical interpretation of relativity: the speech Die Relativitätstheorie im erkenntnistheoretischen Zusammenhange des relativistischen Positivismus (The relativity theory in epistemological connection with relativistic positivism, 1912a); the entry Naturwissenschaften (Natural sciences), for the encyclopedic work Handwörterbuch der Naturwissenschaften (1912b); and the already mentioned paper praised by Einstein, Die Relativitätstheorie der Physik (The relativity theory of physics, 1914). Moreover, thanks to his participation in the scientific circles of the German capital, Petzoldt could keep a close eye on the latest developments of Einstein’s theories. In 1913, he wrote to Mach about his conversations with the physicist Max von Laue, who informed him that Einstein was preparing a new work on gravity, building on Mach’s idea that the centrifugal forces depend on relative rotation (Petzoldt 1913; Blackmore and Hentschel 1985, 120). After receiving Petzoldt’s letter, Mach contacted Einstein again, and sent him the latest edition of the Science of Mechanics. In return, Einstein sent the just published Entwurf of the theory of general relativity, accompanied by these words: Highly esteemed Colleague, you have probably received a few days ago my new paper on relativity and gravitation, which is now finally completed after unceasing toil and tormenting doubts. Next year, during the solar eclipse, we shall learn whether light rays are deflected by the sun, or in other words, whether the underlying fundamental assumption of the equivalence of the acceleration of the reference system, on the one hand, and the gravitational field, on the other hand, is really correct. If yes, then – in spite of Planck’s unjustified criticism – your brilliant investigations on the foundations of mechanics will have received a splendid confirmation. For it follows of necessity that inertia has its origin in some kind of interaction of the bodies, exactly in accordance with your argument about Newton’s bucket experiment (Einstein [1913] 1995, 340).

When Einstein moved to Berlin, he not only recommended Petzoldt’s paper to the readers of the Vossische Zeitung, but also contacted Petzoldt directly. In his letter, Einstein wrote: Highly esteemed Colleague, I read your comments on relativity theory in the Zeitschrift für positivistische Philosophie with much pleasure. From it I see with astonishment that you are closer to me in your understanding of the subject, as well as with regard to the sources from which you draw your scientific convictions, than my true colleagues in the field, even as far as they are unconditional supporters of relativity theory. After great exertion I have now succeeded in establishing proof that the gravitation equations formed last year have a very high degree of covariance with acceleration transformations. Seen from the physical standpoint, rotation and acceleration prove to be entirely relative; there is no distinction between a “real” gravitational field and an “apparent” gravitational field produced through the acceleration of the reference system. In both fields the same field equations of gravitation apply. The only point in regard to which I do not agree with your representations is the matter of the moving clock. [ . . . ] I would be very pleased if we were to see each other one day soon so that we can discuss this question of common interest to us both (Einstein 1914 [1998], 12).

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Even though Einstein does not explicitly mention the points of contact between him and Petzoldt, the hint to the “real” and “apparent” gravitational field indicates that he shared Petzoldt’s interpretation of relativity, according to which the theory is a further step towards the goal of relativistic positivism: that of overcoming the dichotomy between reality and appearance. The end of Einstein’s letter was not just a ritual formula. Indeed, he visited Petzoldt in June 1914, accompanied by his friend and physicist Paul Ehrenfest. On this occasion, Petzoldt gave Einstein a copy of his book Das Weltproblem vom Standpunkte des Relativistischen Positivismus aus (The problem of the world from the point of view of relativistic positivism). Soon after, Einstein sent Petzoldt a reply, in which he wrote: “I just finished reading your book with great interest, from which I gather with delight that I have long shared your convictions” (Einstein [1914] 1998, 24). In addition to these words of appreciation, Einstein also tried to help Petzoldt’s academic career. In a 1914 letter to his cousin and future wife Elsa, we read: “Concerning Petzoldt, I have sent a very warmly phrased letter of recommendation to the minister. It will not misfire” (Einstein [1914] 1998, 40). Then, in 1918, Einstein accepted the request of the mathematician Georg Helm to write a recommendation letter to support Petzoldt’s application for the philosophy chair at the University of Dresden (cf. Einstein [1918] 1998, 511; [1919] 2004, 76). It should be noted that in the time between these two (failed) attempts to help Petzoldt’s career, Mach had died. On this occasion, Einstein wrote a eulogy for the journal Physikalische Zeitschrift, in which he once again emphasized Mach’s “greatest influence upon the epistemological orientation of natural scientists” (Einstein [1916a] 1997, 141). In particular, Mach did not cease to remind us that “concepts that have proven useful in ordering things can easily attain an authority over us such that we forget their worldly origin and take them as immutably given”, so much so that scientists begin to treat these concepts as that scientists start treating these concepts as “most sacred treasures”, and philosophers try to protect them by putting them in the “treasure chest of ‘absolutes’ and ‘a prioris’” (Einstein [1916a] 1997, 142). Since “space” and “time” are such concepts that one might be tempted to hypostatize, with his historical-critical analysis Mach “paved the road for progress” and “helped” Einstein develop the theory of relatively “both directly and indirectly” (Einstein [1916a] 1997, 143). What is more, in the eulogy Einstein even claimed that “it is not improbable that Mach would have hit on relativity theory if in his time – when he was in fresh and youthful spirit – physicists would have been stirred by the question of the meaning of the constancy of the speed of light” (Einstein [1916a] 1997, 144). Before ending, Einstein also touched on the delicate issue of Mach’s alleged sensualism, stressing that his “desire to find a point of view from which the various branches of science [ . . . ] can be seen as an integrated endeavor” led him to regard “all science as a striving for order among individual elementary experiences which he called ‘sensations’”, and because of “this choice of word” “those who are less familiar with his works often mistook the sober and careful thinker for a philosophical idealist and solipsist” (Einstein [1916a] 1997, 145).

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Even after Mach’s death, Petzoldt and Einstein remained in contact. Petzoldt attended Einstein’s conferences in the capital. They probably met in the cultural and scientific circles of the city. And we know that they still visited each other in 1919, since we have a letter in which Einstein invites Petzoldt to his home to explain Ehrenfest’s paradox of the rotation of a rigid disk to him (Einstein [1919] 2004, 74–75, 77). Then, in 1920, we have the last trace of a contact between the two. After the successful experiment during the solar eclipse observation of 1919, more and more philosophers became interested in Einstein’s relativity. Hence, the Kant-Gesellschaft organized a conference in Halle on 29 May 1920 to discuss the new theory, and invited Einstein to attend the event. Even though Einstein initially accepted, he later declined after psychologist Max Wertheimer wrote to him to warn him against participating in such an event. According to Wertheimer, no “physicists of commensurate caliber” should get involved in a discussion with “feeble-minded, languidly regurgitating, squabbling mediocre” philosophers, such as Einstein critic Oskar Kraus, who was scheduled to lecture at the conference (Einstein 2006, 161–62; on the topic see Hentschel 1990b, 168–77). On the other hand, Petzoldt attended the event and ended up acting as the main advocate of Einstein’s theories. For this reason, on his return he wrote to Einstein to inform him of the confusion among philosophers regarding the actual content of the relativity theory and its experimental basis. According to Petzoldt, “full clarity on all these problems will only be possible if physicists and philosophers discuss them together”. Therefore, in the letter, he tests Einstein’s willingness to participate in a new event featuring “relativity theorists interested in philosophy”, rather than philosophers in the narrower sense (Einstein 2006, 205–6). We do not know whether Petzoldt succeeded in organizing the meeting. What is certain is that this is the last letter in their correspondence. Maybe Einstein’s worldwide success, which led him to travel around the world, distanced him from Petzoldt and a milieu of philosophers who were often more interested in using the theory of relativity for their own purposes, than in understanding the real content of the innovations introduced by Einstein. However, we cannot rule out the possibility that the relationship between Einstein and Petzoldt was also adversely affected by the publication of Mach’s The Principles of Physical Optics in 1921. As is well known, this posthumous work included a Preface in which Mach apparently dismissed the theory of relativity: I gather from the publications which have reached me, and especially from my correspondence, that I am gradually becoming regarded as the forerunner of relativity. I am able even now to picture approximately what my new expositions and interpretations many of the ideas expressed in my book on Mechanics will receive in the future from the point of view of relativity. [ . . . ] I must, however, as assuredly disclaim to be a forerunner of the relativists as I withhold from the atomistic belief of the present day. The reason why, and the extent to which, I discredit the present-day relativity theory, which I find to be growing more and more dogmatical, together with the particular reasons which have led me to such a view – the considerations based on, the physiology of the senses, the theoretical ideas, and above all the conceptions resulting from my experiments, must remain to be treated in the sequel (Mach [1921] 1926, vii–viii).

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Thanks to Gereon Wolters’ historical and philological investigations, we now know that these words are an apocryphal text written by Mach’s son Ludwig (Wolters 1987). However, Einstein and his contemporaries had no reason to doubt the veracity of the text. Hence, Mach’s disavowal of the theory of relativity became sensational news in the scientific and philosophical world, all the more so because, at the time, Mach was universally recognized as one of the main inspirations for Einstein’s work. Consequently, it is not difficult to imagine Einstein’s disappointment when he discovered that the thinker who had helped him more than anyone else in shaping his epistemological conceptions unexpectedly turned against him. For this reason, in the following years, Einstein started to disown Mach’s influence on his work. For example, during a conference in Paris in 1922, Einstein answered to a question about his debt to Mach as follows: From the logical point of view there does not seem to be much relation between the theory of relativity and Mach’s theory. For Mach there are two points to distinguish: on one hand, there are things that we cannot budge: these are the immediate facts of experience on the other hand, these are concepts that we can, on the contrary, modify. Mach’s system studies the relations existing between the facts of experience the ensemble of these relations, for Mach, is science. This is a false standpoint here all in all, what Mach had made was a catalog, not a system. As good as Mach was as a mechanician, he was a deplorable philosopher. His shortsightedness about science led him to reject the existence of atoms (Einstein [1922] 2012, 130–31 emphasis mine).

A great change in tone, indeed, compared to the words Einstein wrote in Mach’s eulogy! In light of the above, we may say that the apocryphal Preface written by Ludwig Mach shattered overnight Petzoldt’s years-long efforts to weave together the theory of relativity and Mach’s philosophy. Nonetheless, Petzoldt continued to sustain that the positivistic relativism he developed on the basis of Mach’s ideas was the true epistemological framework for the theory of relativity. Hence, he tried to downplay the Preface of the Optics by stating that Mach’s “pronouncements not favorable towards the ‘present’ theory of relativity” should not obscure the work he had done in the Science of Mechanics against the notion of absolute space and absolute time, which paved the way for Einstein (Petzoldt 1924, 143n). Having summed up the relationship between Einstein and Petzoldt, we can now move on to analyze the latter’s interpretation of the theory of relativity. According to Petzoldt, the significance of the theory of relativity is the fact that it lies on the “main line of the current evolution of human thought”, which consists in the progressive overcoming of the metaphysical search for the substance and in the “elimination of the opposition between ‘real and ‘apparent’” (Petzoldt 1914, 2–3). In particular, Petzoldt distinguishes two phases in the theory of relativity, personified by Mach and Einstein. Building on his studies in the history of physics and in the physiology of the senses, as well as on epistemological reflections, Mach provided a first relativization of space-time and of the shapes of bodies. On the one hand, as we have seen, Mach’s criticism of Newton’s account of inertial and gravitational forces led to a rejection of the notions of absolute space and absolute time. On the other hand, Mach’s psychophysiological investigations brought him to question the overvaluation of the sense

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of touch that was implicit in the mechanistic-materialistic notion of matter. Thus, Mach refuted the idea that the true shape of a body is provided by the matter of which that body is made, which corresponds to what can be touched. As he wrote: But where is a body? Is it only where we touch it? Let us invert the matter: a body is where it acts. A little space is taken for touching, a greater for hearing, and a still greater for seeing. How did it come about that the sense of touch alone dictates to us where a body is? (Mach [1872] 1911, 56).

As a result, Mach rebutted the assumption that the goal of science is to know the supposed absolute reality that is beyond our sensory experience, a reality in which matter and energy, located in absolute space and time, play the role of the old metaphysical substance. Instead of pretending to discover “the world in itself, without any relation to the organization of the one who wants to know”, Mach showed us that the aim of science should be “to assess the relations between man and his near and distant surroundings, to form of a system of conceptual reactions which establishes an equilibrium, a stable relation, a permanent connection, between man and all incoming complexes of stimuli” (Petzoldt 1914, 6). According to Petzoldt, the reason why Einstein was able to find a way out of the problems physicists faced at the time – such as the apparent lack of invariance of Maxwell’s equations, or the contradiction between Fizeau’s, and Michelson and Morley’s experiments (Cassini and Levinas 2018; Norton 2004, 2014) – was that he observed these problems from the point of view of Machian epistemology. Thanks to Mach, Einstein did not view these problems as inexplicable inconsistencies in the world of physics, but rather “simply and naturally accepted them as a new confirmation of his relativistic belief” (Petzoldt 1914, 9). In particular, Petzoldt stresses that the difference between Einstein’s Machian relativism and the (maybe unwitting) metaphysical materialism of the other physicists is evident in their different interpretations of Lorentz’s contraction. When Hendrik Lorentz first proposed to introduce a length contraction to resolve the apparent contradictions of Michelson and Morley’s experiment (which showed that the speed of light remained the same regardless of motion, contrary to Galilean relativity), he assumed that such a contraction was a concrete physical effect caused by the motion of the object in absolute space, determined by the ether resistance. Consequently, for Lorentz, the object was “really” contracted, whereas the constancy of the speed of light was only “apparent”. Conversely, Einstein assumed from the outset that physics cannot and should not deal with absolute motion or the supposedly “real” size and shape of a body. Since Mach taught him that all physicists can do is establish lawful relations between empirical data, Einstein simply acknowledged the relation between the length measurement and the relative motion of the body. Following this Machian approach, for Einstein it made no sense to ask what the actual length of the body was, or whether the body was undergoing a real contraction. Simply, a body can have different lengths depending on the frame of reference, and there is no absolute or preferential frame of reference that gives us the real measurement of the body (Petzoldt 1914, 9 ss.; on the topic see also Miller 1997).

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According to Petzoldt, physicists only need to establish the univocal determination between length and relative velocity. When they try to explain why length varies (i.e. what happens to the matter that constitutes the contracted body), they fall back on the old absolutistic-materialistic worldview. Hence, Petzoldt attached particular importance to Minkowski’s assertion that length contraction should be regarded as a “gift from above”, i.e. as an “accompanying circumstance of the circumstance of motion” (Minkowski [1908] 2010, xxiii; quoted in Petzoldt 1914, 16). For metaphysical physics, which regards everything in terms of material bodies and forces acting on them, such a “gift from above” is impossible: if the length changes, there must be a force that caused the contraction of the body. On the other hand, from a Machian point of view, there is nothing mysterious in this “gift”, it is just an example of the functional relations between the phenomena. Therefore, Petzoldt affirms that, from this perspective, length contraction is not only “unproblematic”, but even “self-evident” (Petzoldt 1914, 17). The rebuttal of the opposition between the “real” world of matter and forces on the one hand, and what “appears” from various perspectives or frames of reference on the other hand, was not the only commonality between the relativistic positivism of Machian origin and Einstein’s theory of relativity. It is not enough to say that each observer has a different picture of the world; that different frames of reference give different measurements for the same object or event; and that they are all equally real because there is no absolute account of the world from a preferential point of view (or from no point of view at all). If we stop here, then relativism leads to skepticism and chaos. But we should not stop here, because the important thing is that there is univocal determination between all the different perspectives, between all the different frames of reference, so that we know what the world must be like from the other points of view. In the case of the theory of relativity, this univocal determination is provided by Lorentz’s transformations, which allow us to calculate what is measured from different frames of reference. Thanks to these transformations, we can translate the measurements obtained in a frame of reference into the measurements we would obtain in a different frame of reference. In Mach’s terms, or – more precisely – in Petzoldt’s terms, Lorentz’s transformations are the “functional relations” that univocally connect all the frames of reference, hence assuring the necessity of the world. The significance of these equations is that thanks to them each point of a system of coordinates can be univocally correlated to a to point of a system moving at constant linear velocity, or that the space of one system can be “mapped” [“abgebildet”] from the space of the other. [ . . . ] Above these spaces there is nothing, and there is no nature beyond them. Each individual observer uses his own space for the goal of his own representation, of his own description of nature, and he acknowledges the corresponding space of every other observer as equally legitimate, and establishes a univocal relation between his representation and the representation of the other man, and maps the representation of the other man on the basis of his own space (Petzoldt 1914, 21).

As it is evident, Petzoldt believes that the ultimate significance of the theory of relativity is no different from that of perspective geometry: a body is different depending on the standpoint from which we look at it; nevertheless there are laws

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that allow us to shift from one standpoint to the other (see Petzoldt 1914, 17). For this reason, Petzoldt disagrees with those who accused the theory of relativity of being unintuitive, unenvisionable (unanschaulich). As perspective geometry shows, we are accustomed to envision a body as having different shapes at once from different points of view. What is unenvisionable is simply the pretension to attribute all these shapes at once to the body in itself, by erasing the reference to the various points of view. The same applies to the theory of relativity: there is no problem in assuming that a body may be simultaneously contracted and not contracted relatively to different frames of reference. Problems arise only when we maintain the old absolutistic approach and try to ask how the body in itself can be contracted and not contracted at the same time (Petzoldt 1914, 31). We should stress that the issue whether the theory of relativity is envisionable or unenvisionable is not only about its understandability, but is also about the epistemological status of scientific theories. Following Mach, Petzoldt believes that a scientific theory, no matter how abstract, is always based on the sensory experience and is always an abridged description of the sensory experience. Hence, Petzoldt rails against the idealist tradition in philosophy – beginning with Plato and extending to Kant and his successors – because it regards sensory experience as a sheer appearance and wrongly insists on establishing scientific knowledge upon a-priori pure rational categories. In Petzoldt’s own words: Those who regard the senses as deceivers deny the value of the natural sciences for the knowledge of reality, since there is no valid natural-scientific statement that is not based upon sensory observation. [ . . . ] The distrust of the senses is still today the basis of the outrageous doctrine of epistemological idealism, according to which the world is just a representation, an appearance in consciousness. Thus, some theorists of relativity follow the same path with their doctrine about the “appearance”. Einstein correctly recognized that the question whether or not Lorentz’s contraction actually exists can be misleading. “It does not exist ‘in reality’ inasmuch as it does not exist for a moving observer; but it does exist ‘in reality’, i.e., in such a way that, in principle, it could be detected by physical means, for a noncomoving observer”. In this is already included that for the noncomoving observer the contraction is not just an “appearance”, and thus the rights of the senses are re-established also by modern physics – something that in ancient times Protagoras had attempted in vain. [ . . . ] We are at the border between two Weltanschauungen. It is not possible to invoke the theory of relativity to support both. This theory has emerged from the circle of ideas of relativistic positivism and it can only support the latter (Petzoldt 1914, 42–43; it contains a quote from Einstein [1911] 1994).

As we can see, Petzoldt emphasizes the sensory-empirical character of the theory of relativity to support the thesis that there is an inherent connection between the radical empiricism advocated by Mach and himself, and relativity theory. This move becomes even more evident in his later writings on the theory of relativity. In 1916 Einstein introduced the concept of “coincidences”, stating that “all our spacetime verifications invariably amount to a determination of space-time coincidences” (Einstein [1916b] 1997, 153; on the topic see Giovanelli 2021). In a later book, Petzoldt adopted Einstein’s new concept in his own way, claiming that all scientific data are “observations and judgments concerning complexes of sensations or, at best, complexes of memories of sensations: Einstein calls them coincidences, we

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should call them coincidences of sensations” (Petzoldt 1923, 37). In so doing, Petzoldt stretched Einstein’s notion to bring it closer to Mach’s assertion that “every physical concept is nothing but a certain definite connexion of the sensory elements”(Mach [1886] 1897, 192). Accordingly, Petzoldt reformulated the theory of relativity as follows: The observers ascertain that their simple coincidences of sensations (their observations of meter sticks and clocks) diverge from each other, but that there is full accordance in their pair of coincidences (in the “natural laws”). Thereby, these divergences are univocally determined (thanks to Lorentz’s transformation) and therefore are, in their turn, governed by natural laws. Hence, even these “simple” coincidences ultimately are pairs or complexes of coincidences, since they must be correlated to the velocity of the systems and also to the corresponding configurations of the sensory nerves and the central nervous structures (Petzoldt 1923, 73–74 emphasis mine).

This passage clearly shows how Petzoldt justifies the shift from the coincidences in the Einsteinian sense to his coincidences of sensations. For Petzoldt, there are no absolute, irrelated data in knowledge; each datum exists only in a series of relations with other data. However, among these data there is also the human brain, whose variations are the ultimate condition of our observations. Consequently, for Petzoldt even Einstein’s coincidences are in their turn dependent upon the brain; they are coincidences with certain cerebral states, and in this sense they should be called coincidences of sensations. However, it is important to stress that the emphasis Petzoldt puts on the observer and on sensations does not mean that he attributes a mentalistic or subjectivistic character to the theory. We must not forget that, according to Petzoldt, sensations are nothing but functional relations between what is in the world and brain activity. Hence, they are an example of the countless functional relations investigated by physics, rather than mere appearances in the mysterious inner world. For this reason, Petzoldt stresses that “there is no reason to fear that with the plurality of observers a foreign psychological moment ( . . . ) would be introduced into physics”, since observer means merely a “connection of systems of reference and systems of coordinates” (Petzoldt 1923, 67). In other words, for Petzoldt talking about “coincidences of sensations” is a way of remembering that the brain is the system of reference for all physics, or – more generally – for all knowledge, since we do not and cannot know anything absolute, nothing beyond the relation between our brain and the world. In this respect, the common feature of Petzoldt’s Machian philosophy and Einstein’s theory of relativity is that both acknowledge that any description of a phenomenon is always given from a specific frame of reference, be it a fourdimensional system of coordinates or the nervous system. Therefore, they both reject the idea typical of materialistic mechanism that there is a single description of nature, and they both affirm instead that the only true image of the world is the multiplicity of all the descriptions and the univocal determination that links them all.

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2.5 Conclusions From our investigation it is clear that, even though Petzoldt developed his philosophy starting from Mach’s ideas, their positions did not entirely overlap. Consequently, when speaking of Einstein’s ties to the Machian Denkkollectiv, one should always distinguish two issues: on the one hand, the influence that Mach exerted on Einstein; on the other hand, the quasi-Machian interpretation of the theory of relativity proposed by Petzoldt and (partially and initially) endorsed by Einstein himself. Since Mach did not engage directly in the debate over relativity, and since Petzoldt’s opinions about the relativity of space, time, and motion differ from Mach’s, it is impossible to get an accurate picture of Einstein’s links with Machian philosophy without including Petzoldt. Mach surely influenced Einstein with his criticism of Newtonian physics, his research program of a unified theory of gravity and inertia, and – more generally – his historical-critical approach, that strips physical concepts of their monolithic semblance. However, there is also a double link that connects Einstein directly to Petzoldt: (1) the interpretation of the theory of relativity as overcoming of the dichotomy of reality and appearance; and (2) the notion of univocal determination that preserves the lawfulness of the connection between all the different frames of reference, thus ensuring the complete knowability of the world. These two points are also the cornerstones of relativistic positivism since they mean, respectively: (1) that there is not the world-in-itself on the one hand, and its sheer appearances constituted by its relations to other things on the other hand, but everything is what it is in relations to other things (relativism); and (2) that science – as the investigation of the univocal determination of these relations – is not only possible, but it is a true knowledge of the world (positivism). This is not to say that Einstein adopted Petzoldt’s relativistic positivism in its entirety; rather that, at least for a while, he thought that the core of this philosophical position could provide a valid interpretation for the theory of relativity. This was partly due to the Machian origin of Petzoldt’s conception, which resonated with Einstein’s own proximity to Mach’s ideas; but also because Petzoldt’s notion of Eindeutigkeit secured a stronger foundation for physics than Mach’s indeterministic tendency of Humean influence. Without neglecting Mach’s emphasis on the biological and pragmatic basis of knowledge, Petzoldt always insisted that the world is governed by univocal determination and that we can therefore have a positive knowledge of the world, because we ourselves are connected to it through univocal determination. Such a Machian philosophy revised in favor of a stronger determinism must have appealed to Einstein, who always fought against the idea that phenomena could happen without being completely determined, as would become more and more evident with the advent of quantum mechanics. As Einstein wrote in 1924 in a letter to Max Born: I won’t be driven into abandoning strict causality before entirely different defenses have been tried against it than hitherto. The thought that an electron subjected to a ray chooses of its free volition the instant and direction in which it wants to bounce away is intolerable to me. If so, then I would rather be a cobbler or even a casino employee than a physicist. (Einstein 2015, 237).

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In sum, we may say that the “interpretational frame” provided by Petzoldt was surely built on Machian grounds (such as the refutation of the metaphysical materialistic mechanism implicit in most physical investigations), but also contained a non-Machian element in the principle of univocalness, that was in contrast to Mach’s indeterminism. What matters is that both these Machian grounds and this non-Machian element appealed to Einstein, and led to his favorable reception of Petzoldt’s interpretation of relativity theory during the 1910s.

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Chapter 3

The End of Matter? On the Early Reception of Relativity in neo-Kantian Philosophy Paolo Pecere

Abstract In his article La fin de la matière (1906) Henri Poincaré reported that according to many physicists “matter does not exist”, but he immediately added: “this discovery is not conclusive”. This caution was not shared by many philosophers, who swiftly saluted both special and general relativity as the sources of a new conception of physical objects. In my talk I will focus on Marburg neo-Kantianism (Cohen, Natorp and Cassirer) with its characteristic thesis of a progressive “dissolution” of matter modern physics, culminating in relativity. I will point out that this view predated Einstein’s discoveries and was rooted in a long epistemological tradition starting with Leibniz and Kant. Indeed, the neo-Kantian philosophers argued that this view was an originally Platonic insight that modern physics (and Einstein’s relativity in particular) eventually confirmed. I will show that this historical-philosophical perspective influenced the reception of relativity and left open a number of epistemological and scientific problems.

3.1 Introduction: Einstein’s Relativity, Matter and Field The hypothesis that relativity theory entailed the rejection of the concept of matter was spelled out by Einstein in a number of places. The statements in the popular book “The evolution of physics”, written with Leopold Infeld, are quite clear in this regard. We cannot build physics on the basis of the matter-concept alone. But the division into matter and field is, after the recognition of the equivalence of mass and energy, something artificial and not clearly defined. Could we not reject the concept of matter and build a pure field physics? What impresses our senses as matter is really a great concentration of energy into a comparatively small space. We could regard matter as the regions in space where the field is extremely strong. In this way a new philosophical background could be created

P. Pecere () University of Roma Tre, Rome, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Russo Krauss, L. Laino (eds.), Philosophers and Einstein’s Relativity, Boston Studies in the Philosophy and History of Science 342, https://doi.org/10.1007/978-3-031-36498-3_3

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P. Pecere [ . . . ] There would be no place, in our new physics, for both field and matter, field being the only reality. (Einstein and Infeld 1938/1967, 257)

The “new philosophical background” was presented as a possible implication of physical discoveries. Notably, Einstein’s relativity theory (both special and general) was characterized as merely a part of this process. Indeed, general relativity did not entirely realize Einstein’s idea of a “new physics”; it rather posited a new problem: This new view is suggested by the great achievements of field physics, by our success in expressing the laws of electricity, magnetism, gravitation in the form of structure laws, and finally by the equivalence of mass and energy. Our ultimate problem would be to modify our field laws in such a way that they would not break down for regions in which the energy is enormously concentrated. (Ivi, 258)

Thus Einstein introduced relativity in a series of physical theories of field that culminated in the “programme” of a “pure field physics”. The two achievements that suggested the need for a new breakthrough were the development of “structure laws” in classical and relativistic field theories and the “equivalence” of mass and energy. This suggested that mass – apparently coextensive with “matter” – might be also reduced to a property of a field in a new physical theory. As is well known, Einstein constantly believed that the continuity of fields and the discreteness of particles produced a “provisional and unsatisfactory” dualism in post-relativistic physics. After the rise of quantum mechanics produced a quite different background for the physics of matter, Einstein maintained that the need for a new field theory was not removed, and always considered quantum mechanics as a provisional theory. In a letter to Michele Besso of August 10, 1952 he still argued for the need of a theory where particles are “deduced” rather than postulated (Einstein and Besso 1972, doc. n. 190). He worked until his last years on a pure field theory that would present particles (whether point-like or extended) as solutions of fields. However, Einstein never managed to realize this programme.1 Besides the open status of Einstein’s late research programme, the interpretation and assessment of his views on field and matter are complicated by a number of further reasons. First, as it has been pointed out, Einstein’s apparent identification of matter and mass in the above quoted passages, as well as his occasional talk of the substantial identity of mass and energy,2 were problematic – although widely accepted among early interpreters – and still raise substantial interpretative issues.3 Second, contemporary studies of “structure laws” in early twentieth century physics are focused on different contributions, such as the “structural” epistemology of Poincaré and Cassirer or the programs of a unified field physics by Hermann

1 On

these research see Pais 1982, 312–313, 345–376, 488–497. Also see Einstein’s own account in Schilpp 1949, in part, 71–94. 2 “Mass and energy are essentially alike. They are only different expressions for the same thing” (Einstein 1953, 45). 3 For an outline of this problem, with useful historical remarks, see Lange 2002, ch. 6. Lange also quotes prominent scientists who accepted at face value Einstein’s identification of mass and energy, e.g. Werner Heisenberg.

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Weyl and Arthur Eddington; then the application of group theory and the rise of quantum field theory have provided a different scientific background for successive investigations of these issues.4 Indeed, assessments of “structuralism” and “structural realism” in contemporary philosophy of science are more indebted to these accomplishment than to Einstein’s original theories, and Einstein himself is not considered a “structuralist” because he always believed in the existence of selfsubsistent, independent physical objects (the deduction of discrete parts of matter from continuous fields being a different issue from the identification of objectivity in physics with pure structural relations).5 On the whole, to consider Einstein’s original views on field and matter in both a historical and theoretical perspective is hardly straightforward. In this paper, I will not deal with this problem. I will rather deal with early neo-Kantian interpretations of relativity, focusing on the way Cassirer contextualized Einstein’s theories in the historical narrative of modern science. I will start from two observations made by Cassirer: first, he correctly pointed out that a number of physical and philosophical theories had made the hypothesis that matter might be conceived as a product of a different dynamical concept (force, field, or energy) before the formulation of relativity, and thus provided a background for Einstein’s programme of a post-relativistic field theory; second, Cassirer also pointed out that Weyl’s and Eddington’s interpretations of relativity dealt with the hypothesis of mass as a product of field (Sect. 3.2). In the light of this series of promising theories, Cassirer considered relativity as conducive to “pure field theory”. At the same time, he also conceived this succession of theories as the confirmation of the hypothesis formulated by his teacher Hermann Cohen of a “dissolution of matter”, as a progressive result of the history of physics (Sects. 3.3 and 3.4). However, as I will argue, this hypothesis was motivated by a “Platonic”, neo-Kantian epistemological programme that was foreign to Einstein’s original views (Sect. 3.5). Cassirer subtly recognized that this programme agreed with a trend in relativistic theories of matter. However – as I will argue – he did not clearly recognize the open problems in these theories, which produced a substantial flaw in the epistemological project of Marburg neo-Kantianism, casting doubts on the conclusion that relativity prepared the “victory of idealism”.

4 For

a history of these investigations in early relativistic physics (especially on Weyl and Eddington) see Ryckman 2005. 5 “Structural realism”, as the theory – or family of theories – that only the mathematical or structural content of scientific theories can be considered as their real ontological content, has been debated as an important version of scientific realism since at least the 1970s (Maxwell 1970; Worrall 1989; Cf. Ladyman 2020). However, early models of what is now called structural realism date back to the early twentieth century. Cassirer, as we will see, played a major role in this early story (see Gower 2000). On the metaphysics of structural realism and Einstein’s belief in individuals see Ladyman and Ross (2010), 148–154, in part. 151.

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3.2 Cassirer on Relativity and “Field Physics” The idea of a physics without matter, formulated by Einstein in the above quoted account, predated the formulation of relativity theory. In the late nineteenth century, the electromagnetic theory of mass had been debated as a possible means of explaining matter away. In the article “The End of Matter” (originally published in 1906) Henri Poincaré had reported that according to many physicists “matter does not exist”. However the issue was far from settled, and Poncairé immediately added: “this discovery is not conclusive” (Poincaré 1908, 282). The electromagnetic theory of mass never gained the status of a confirmed hypothesis.6 It is no surprise that it was not even mentioned in the popular account of physical theories by Einstein and Infeld. Hence it might be argued that Einstein explored a new way of realizing the explanation of mass in terms of fields and thus fulfill the aim of the electromagnetic theory of mass. Nevertheless, the origin of his programme cannot be identified with this theory. The very idea of a dynamical explanation of mass predated the electromagnetic theory of mass as well, and Einstein himself, after the publication of special relativity, was familiar with a number of epistemological and scientific theories that mentioned this broader context. A prominent example was Cassirer’s neo-Kantian “critique of knowledge”. In his book Einstein’s Theory of Relativity (1921), Cassirer argued that general relativity was a “natural logical conclusion of an intellectual tendency characteristic of all the philosophical and scientific thought of the modern age” (ECW 10, 34).7 According to Cassirer, Einstein’s theories demonstrated – against positivism – the active role of “physical thought” in the constitution of scientific objectivity: The new concept of nature and of the object, which the theory of relativity establishes, is grounded in the form of physical thought and only brings this form to a final conclusion and clarity. Physical thought strives to determine and to express in pure objectivity merely the natural object, but thereby expresses itself, its own law and its own principle. (ECW 10, 111)

Thereby relativity confirmed the value of Kantian insights, and in particular of Cassirer’s original elaboration of neo-Kantianism in Substance and Function (1910), where he had argued that scientific thought in its historical development tends to reject the very idea – suggested by the intuitive representation of objects – that truth is correspondence of theories with the properties of a substance and leads towards a conception of knowledge as a functional “coordination” of phenomena according to laws. In this sense relativity was the “triumph of the critical concept of function over the naive representation of the thing and the substance” (ECW 10, 65). In particular, special relativity showed the dependence of the form (Gestalt) of bodies from the reference system and thus it ruled out the similarity between

6 Harman 7I

1982, 72–119, 149–155. Cf. Jammer 1997, 136–153. quote the English translation in Cassirer 1923.

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“optical pictures” and “absolute form [Form] of the object” (ECW 10, 50). General relativity eventually disposed of the substantialist meaning of mass. The geometrical interpretation of mass destroyed the foundations of the two main hypostatical concepts of classical physics, atom and ether. First, the new theory of gravitation removed the “dualism” of space and matter that characterized atomism from the Antiquity to Newton (ECW 10, 53–57). Second, the principle of equivalence of inertial and gravitational mass removed the separation of matter and force of Newtonian mechanics, that had formed the basis of a dynamics that could not deduce mass and the laws of motion from the same principle. Finally, with the unification of the laws of the conservation of mass and energy, general relativity realized the epistemological objective that eighteenth century energetics had postulated but – since it considered inertial mass as an independent variable – only realized in a “logically unsatisfactory” way (ECW 10, 10, 57–63). In general relativity the intuitive representation of matter was finally abandoned, as was confirmed by the fact that the new expression of the line element by means of Gaussian coordinates did not include any reference to the “fixed and rigid reference body”. Here was expressed again “the characteristic procedure of the general theory of relativity; while it destroys the thing-form [Dingform] of the finite and rigid reference body it would thereby only press toward to a higher form of object, to the true systematic form [Systemform] of nature and its laws” (ECW 10, 62). Thus general relativity represented a new step in the process of “objectification”, since “by the nature of physical thought, all its knowledge of objects con consist in nothing save knowledge of objective relations”. In other words, relativity confirmed the structural and relational view of scientific knowledge that had been recently advocated by prominent mathematicians and physicists, e.g. Poincaré and Russell, but whose philosophical formulation was already made in Kant’s transcendental logic. Cassirer found support for the latter claim in a passage from the Critique of Pure Reason: “whatever we can cognize only in matter is pure relations (that which we call their inner determinations is only comparatively internal); but there are among these some self-sufficient and persistent ones, through which a determinate object is given to us” (Kant 1900, III, 229. Cf. ivi, III, 69). Thus Cassirer subtly detected a Kantian root of contemporary structuralism, which, in turn, was confirmed by general relativity, for the latter had “shifted these ‘independent and permanent’ relations to another place by breaking up [in dem sie ( . . . ) auflöste] with both the concept of matter of classical mechanics and the ether of electrodynamics” (ECW 10, 41–2). This progressive narrative culminated in the transformation of matter into field. Cassirer presented Einstein’s results as the new step of a “progressive transformation of the concept of matter” in “field physics”, initiated by early attempts at reducing mass to the electromagnetic field, from Michael Faraday to Gustav Mie and Walter Kauffman (ECW 10, 55–57). This transformation was conceived in the framework of Cassirer’s view of scientific progress, according to which successive theories include and expand the relations among phenomena that are introduced in former theories. Cassirer had elaborated this view of scientific progress in Substance and Function (1910).

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This neo-Kantian interpretation, I submit, reflected Einstein’s original insights only to a limited extent, and was substantially dependent on different scientific sources. To be sure, Einstein was familiar with Kant’s philosophy since his youth and he was aware of the rising neo-Kantianism. Einstein notably attended August Stadler’s lectures on Kant and on “Theory of Scientific Thought” at the Swiss Federal Polytechnic Institute in Zürich in 1897. Stadler was Hermann Cohen’s Phd student and he published a book, Kants Theorie der Materie (1883), where he examined Kant’s dynamical theory of matter in the 1786 Metaphysical Foundations of Natural Science.8 In turn, the new “pure physics” that Kant wanted to establish in this book and recommended mathematical physicists to include in their treatises involves a rejection of Newton’s position of hard particles as originally created by God: mechanical properties of matter (such as mass) were rather based on the action of fundamental attractive and repulsive forces of matter. Kant’s dynamical theory of matter was a standard element of the Marburg neo-Kantian narrative of the transition from perceptual to purely ideal representations of matter, and was also mentioned in Cassirer’s book of relativity. Now, since Einstein was aware of these precedents and he also read Cassirer’s book, scholars have wondered whether he was influenced by Kantianism. Indeed, he shared some broadly Kantian claims. His partial appropriation of Kant was not exceptional in the early twentieth century, as varieties of Kantianism abounded. Einstein himself, in the famous 1922 discussion of relativity theory in Paris, provokingly declared that “every philosopher has its own Kant” (Einstein 1922, 20).9 In the controversy over quantum mechanics he starkly rejected the “positivistic attitude” (of Bohr, Heisenberg and others) that restricted possible knowledge to the observable and, in this context, he highlighted the valuable Kantian insight concerning the importance of concepts for the definition of physical objectivity. The objective factor [in sense impressions] is the totality of such concepts and conceptual relations as are thought of as independent of experience, viz., of perceptions. So long as we move within the thus programmatically fixed sphere of thought we are thinking physically. Insofar as physical thinking justifies itself ( . . . ) by its ability to grasp experiences intellectually, we regard it as ‘knowledge of the real’ [ . . . ] The theoretical attitude here advocated is distinct from that of Kant only by the fact that we do not conceive of the ‘categories’ as unalterable (conditioned by the nature of the understanding) but as (in the logical sense) free conventions. They appear to be a priori only insofar as thinking without the positing of categories and of concepts in general would be impossible as is breathing in a vacuum. (Schilpp 1949, 674)10

8 For

an overview of Einstein’s epistemological views and the scholarly debate on this subject see Howard and Giovanelli (2019). Also see Giovanelli (2003) for an analysis of Stadler’s works. I thank an anonymous reviewer for pressing me to include Stadler in my story. See below Sect. 3.4 for Cohen’s account of Kant’s dynamical theory of matter. 9 As it has been argued by Massimo Ferrari in an insightful essay, the same might be said of every German physicist between the end of the nineteenth century and the early twentieth century (Ferrari 2006, 183). 10 Also see Einstein’s replies to the essays by Hans Reichenbach and Henry Margenau, ivi pp. 676–679. Reichenbach’s theory of a “relativized a priori” in The Theory of Relativity and A Priori Knowledge (1921) might have corroborated Einstein’s views.

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Einstein was also aware of the historically conditioned value of relativity. He maintained that relativistic mechanics did not entirely dismiss Newtonian mechanics, but rather included the latter as a limit case, commenting that “no fairer destiny could be allotted to any physical theory, than that it should of itself point out the way to the introduction of a more comprehensive theory, in which it lives on as a limiting case” (Einstein 1917/1920, 91–92). Cassirer promptly mentioned this thesis, which sounded as a confirmation of his view of the continuity of scientific theories (ECW 10, 17–18, 33). But Einstein was also aware of the open status of his program of field theory of matter. Indeed, Einstein might have considered relativity itself as a step in the progressive series of scientific theories that would be included in a comprehensive field theory. Be that as it may, the construction of that “larger theory” was still in the making. In hindsight, we can see that Cassirer’s interpretation went beyond what Einstein himself had established. Cassirer indeed found substantial support for his view of field theory in the interpretation of relativity developed by Hermann Weyl. Weyl’s Space Time Matter (1918) was a major reference in Cassirer’s book on relativity, especially when he commented on the “new physical view” of general relativity, as a theory that “no longer recognizes space, force and matter as physical objects separated from each other [since] for it exists only the unity of certain functional relations, which are differently designated according to the system of reference in which we express them”. In this context, Cassirer cited Weyl’s striking statement that “all physical phenomena are expressions of the world metrics” (SR 398). In fact, Weyl presented the equation of energy and mass in general relativity as a realization of a “pure dynamical view of matter” According to Weyl, “the theory of fields has to explain why the field is granular in structure and why these energy-knots preserve themselves permanently from energy and momentum in their passage to and fro [ . . . ] therein lies the problem of matter” (Weyl 1952, 208). In other words, the theory of fields had to explain “atoms and electrons” as knots in a field. The solution to this problem lied in the energy tensor of general relativity, resulting in the conclusion that “it is not the field that requires matter as its carrier in order to be able to exist itself, but matter is, on the contrary, an offspring of the field”. Hence, “in the future we shall assign the term matter to that real thing, which is represented by the energy-momentum-tensor” (ibid.). Weyl also sketched a progressive narrative of the history of the problem of matter that included most of the references of Cassirer’s own narrative. He compared his conclusion to Kant’s dynamical theory of matter in the Metaphysical Foundations of Natural Science of 1786, citing Kant’s “doctrine that matter fills space not by its mere existence but in virtue of the repulsive forces of all its parts” (Weyl 1952, 202 n.). This was of course an important precedent for neo-Kantian accounts of the problem as well (see below Sect. 3.4). Weyl also pointed out how the electromagnetic field was originally contrasted to “matter” and drew a line from Faraday and Maxwell to general relativity and his own metrical theory of electromagnetism and gravitation, presenting the latter as a transcendental idealistic

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theory of space and time (Weyl 1952, 2–3).11 Here we easily recognize the pattern of Cassirer’s account of relativistic physics. However, Weyl himself had not completed his program. In a passage that Cassirer did not mention, Weyl recognized that his metrical theory of electromagnetic and gravitational field was still “lying in the deepest obscurity” (Weyl 1952, 311). Moreover, Weyl changed his theory considerably after the first edition of Space Time Matter in order to address criticism of his first formulation of the theory, and eventually he developed a different field theory, connecting quantum mechanical indeterminism with the possibility of a causality grounded on a reality existing beyond the spacetime manifold. On this theory, matter was no longer “dissolved” into the metrical field. Nevertheless, Cassirer did not address these important changes and kept referring indiscriminately to Weyl’s “field theory” as a proof that matter is reduced to field. In the third volume of the Philosophy of Symbolic Forms – the Phenomenology of Knowledge (1929) – Cassirer repeated his reference to Weyl’s early program. He maintained that a full “dissolution” of matter could be found in the postrelativistic “theory of field”. Therefore he celebrated the epistemological meaning of the “transition from the physics of matter to the pure ‘field physics’”, as the latter entailed the transition from the intuitive, “geometrical” schematism of classical mechanic to the “universal schematism of the concept of number”, where physical objectivity was defined without any reference to the “world of intuition in its primitive form”, for the field was indisputably an abstract, mathematical concept, with no connection whatsoever to sensory perception. This kind of view, again, had not been developed by Einstein himself, but rather by early interpreters of relativity, such as Weyl in Space Time Matter and in the Philosophy of Mathematics and Natural Science (1927), Alfred North Whitehead in An Enquiry Concerning the Principles of Natural Knowledge (1919), Arthur Eddington in Space, Time and Gravitation (1920) (see ECW 13, 507, 524–525, 540–541, 547–549).12 In particular, Cassirer echoed Weyl’s presentation of matter as a “product of the field”, as Weyl wrote that “what we define as the ultimate physical reality has lost his character of thing: it makes no sense to talk about the same matter in different times” (Weyl 1952, 202–203. See below Sect. 3.5).13 Nevertheless, all these statements concerning the meaning of “pure field physics” belonged to a moment of physical and philosophical research that did not lead to ultimately solid results. Thus Cassirer defended an overoptimistic and slightly generic picture of “field physics”, which significantly characterized the interpretation of 11 Weyl’s main philosophical model was Husserl’s phenomenology,

but was certainly familiar with neo-Kantianism. On Weyl’s field theory and their philosophical background see Ryckman (2005, 77–94) and Scholz (2006). 12 On the notion of “schematism” and its three stages – perceptive, intuitive or geometrical and numerical, see the account in the manuscript Ziele und Wege der Wirklichkeitserkenntnis (1937), in Cassirer (1995–2021), vol. 2. 13 On Eddington, who also maintained that modern physics had given a “death blow to the [ . . . ] materialistic conception of the ether”, see Ryckman 2005, 108–234.

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relativity. His interpretation of contemporary physics was probably influenced and certainly corroborated by the occasional dogmatic statements of his sources. But Cassirer was a subtle scholar and hardly missed the passages that disconfirmed his convictions; hence we must search for a different explanation of his interpretation. As a matter of fact, the program of a unified field confirmed a postulate on the history of science that Cassirer had formulated since his first works, before the rise of relativity.

3.3 Cassirer’s Historical Epistemology and the Problem of Matter In the book on Leibniz, Cassirer already pointed out that the law of continuity entailed the reduction of the material substance to the law of forces and the rejection of the “crass sensory picture” of Cartesian substance, which resulted from the weakness of our sensory intuition. To be sure, as Cassirer was aware that mathematical physics from Galileo to Newton still admitted the existence of independent massive particles. Indeed, he admitted that a full scientific instantiation of these Leibnizian theses could only be found in nineteenth century physics (ECW 1, 254–71). However, as we have seen, Cassirer admitted that even the concept of energy did not entirely rule out the hypostatical notions that he wanted to see excluded from natural science, hence this result was not realized until “field physics”. Here we find an aspect of Cassirer’s argument that needs to be clarified. When Cassirer maintained that a single doctrine realized the ideal of the “dissolution” of matter, and at the same time recognized that this realization was not complete and was rather contradicted by some element of that very doctrine, he was not contradicting himself. His point was rather that the “critical” perspective of the historian allowed to separate the hypostatical and functional trends in a single historical doctrine. At the same time, when he presented a detailed account of crucial notions of modern physics in Substance and Function, Cassirer claimed that the history of physics turned out to follow an epistemological direction: “matter itself becomes an idea, for it is more and more clearly reduced to the ideal conceptions that are produced and confirmed by mathematics” (ECW 6, 184. Cf. 206). This means that the original notion of matter as a “substance” that is identified with a particular sensory element, which was typical of Aristotelian physics, is gradually eliminated in the progress of physics (ECW 6, 164), although the hypostatical tendency keeps manifesting itself in modern theories, e.g. in the notion of electric fluids and caloric (ECW 6, 166– 167). Indeed, the very concept of the atom was often the object of hypostatical thinking, for it is “the analogue and, as it were, the reduced model of the empirical sensory body” (ECW 6, 218). In fact, the sensory and the intellectual notion of the atom were deeply intertwined from ancient atomism to classical mechanics, although Galileo already formulated the epistemological idea that “the substance

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of the physical body” is resolved into the properties that arithmetic, geometry and mechanics discover into it (ECW 6, 169, 180–184).14 Still a full clarification of this idea required the reduction of physical extension to a more abstract principle, as it happened in Roger Boscovich’s natural philosophy with centers of forces (ECW 6, 172). In his account of nineteenth century atomistic, e.g. in Boltzmann, Cassirer still admitted that the status of matter was controversial, although he identified a constant tendency to abandon the “picture” of the atom (ECW 6, 174). The notion of energy finally entailed an anti-hypostatical principle in itself which provided the ground for the transition from spatial determinations to purely numerical determinations, in that energy unified mechanical, thermal, optical and electromagnetic phenomena by means of a mathematical “principle of coordination”, a “pure relation of reciprocal dependency” of variables that identifies the physical object as a structural invariant with no sensory correspondence (ECW 6, 206, 208–9, 218). But this epistemological interpretation, which was confirmed by the work of scientists such as Robert Mayer, was also historically contradicted by the theorist of “energetics” Wilhelm Ostwald, who defended a hypostatical representation of energy (ECW 6, 150, 151–153; cf. 215). On the whole, Cassirer characterized the concept of matter (in Kantian terms) as an “intellectual schema”, the form of the object in the framework of a peculiar scientific system, rather than a transcendent being. The ultimate determination of matter was a regulative idea which the series of scientific theories approached “more and more” as its “limit structure” (ECW 6, 137–139, 175, 178). In order to connect this epistemological idea with the historical data Cassirer argued, in the “Introduction” of the Problem of Knowledge, that “the very concept of history of science contains the idea of the conservation of a general logical structure in any succession of particular conceptual systems [ . . . ] To be sure, also the idea of an internal continuity is nothing but a hypothesis, that however – like all purely scientific presuppositions – is at the same time a necessary condition of the beginning of historical knowledge. (ECW 2, 13, my italics)

Cassirer made clear that the notions of “conservation”, “continuity” and “necessary condition” in this passage should not suggest a picture of the history of science as a linear and uninterrupted progress. “Critical periods” indicated that scientific progress is in fact no mere “continuous quantitative growth”; there is rather a “dialectical contradiction” between different scientific “insights” [Grundanschauungen], as “a concept earlier considered as contradictory can later become both a means and a necessary condition of knowledge”, while principles that formerly explained phenomena can be rejected as “absurd and unthinkable” (ECW 2, 4). The characteristic resort to dialectic suggests that Hegelian philosophy was an important

14 Galileo,

in the third letter to Welser, also admitted that his science only concerned “some accidents [affezioni]” afor it was “impossible” to grasp the “true and intrinsic essence of natural substances” (Galilei 1968, V, 187). Cassirer’s interpretation had to downplay the importance of such statements.

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model for Cassirer’s project of reconciling theoretical change with the “necessary condition” of continuity in the history of science.15 Indeed, a certain element of teleological history and Hegelianism was part of the historical epistemology of the Marburg school since Cohen (see below Sect. 3.4). Against this background, we can understand better Cassirer’s approach to mass and field in the book on relativity. His account of electromagnetism removed the corpuscular elements in early formulations of the theory (e.g. in Maxwell) and downplayed the open status of the electromagnetic theory of mass.16 Cassirer found an apparent solution of the problem of matter in Walter Kaufmann’s theory of inertial mass, but then admitted that a further step – relativity – was still required. The inertia of matter [ . . . ] seems completely replaced by the inertia of energy; the electron – and thus the material atom as a system of electrons – possesses no material but only ‘electromagnetic’ mass. What was previously regarded as the truly fundamental property of matter, as its substantial kernel, is resolved into the equations of the electro-magnetic field. The theory of relativity goes further in the same direction. (ECW 10, 60)

We recognize here Cassirer’s method: he isolates the epistemological intention of a physical theory from its mathematical formulation and empirical confirmation (and possibly also from the original intention of the theoretical physicists themselves); then he grants that the full theory was still unable to perfectly fulfill the epistemological ideal, and introduces a successive theory that goes a step further in this direction. This suggests that he may have applied the same procedure to relativity: he derived the epistemological meaning of the theory from the consideration of Einstein’s equations, popular writings and programmatic declarations; then he might have noticed that Einstein himself wavered as to the proper reduction of mass to the energy of fields, but he considered the interpretations, theories and programs by Weyl, Eddington and others, where he found formulations that matched his own epistemology. This led to his narrative of “field physics”. I will elaborate on the background of this interpretation in the following paragraphs. At this stage, we can already conclude that Cassirer’s “structural” interpretation of relativistic physics has to be distinguished by similar attempts. He was not motivated by the need to combine a realism concerning scientific entities and theory change, as many contemporary “structural realists”. His objective was also not limited to showing that the Kantian notion of a priori knowledge might be compatible with theory change (as Reichenbach in The Theory of Relativity and A Priori Knowledge of 1920). Cassirer was rather motivated by the need to eliminate any residue of intuitive elements (both sensory and geometrical) from the physical notion of matter. This project of a “dissolution of matter”, in turn, was not motivated by purely epistemological criteria, such as the inconsistency between continuity of

15 Cassirer

(ECW 12, 33) ultimately endorsed Hegel’s idea of philosophy as a teleological path towards self-consciousness. Kim (2015, 48) also points out an Hegelian side of Marburg historiography. 16 For historical reconstructions of the electromagnetic theory of mass see Jammer 1997, 136–153. Cf. Harman 1982, 72–119, 149–155, and, for the connection to Einstein, Pais 1982, 172–177.

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fields and discrete representation of matter that worried Einstein, or by ontological parsimony. It was rooted in a controversy between idealism and materialism that provided one of the signature characteristics of Marburg idealism.

3.4 Platonism in Marburg: “Dissolution of Matter” and Relativity as “The Victory of Idealism” As we have seen, Cassirer often refers to a “dissolution” (Auflösung) of the concept of matter, meaning a reduction of this notion to mathematical relations, which removes all sensory elements from the representation of objectivity. This idea belonged to the philosophical program sketched by his teacher Hermann Cohen, which entailed an epistemological reappraisal of Platonism across the history of science. This program, in turn, was supported by an interpretation of Plato’s theory of ideas as a hypothetical and mathematical method, which provided the background for modern science.17 According to Cohen, Platonic ideas play an epistemological role mainly as mathematical ideas. In fact, sensory perception is merely the “stimulus” for “pure mathematical thought”, which investigates the true properties of objects (Cohen 1878, 17–8). Hence existence does not belong properly to sensory objects but only to what is defined by the “methodical connection of thoughts” (ivi, 27). This interpretation of Plato’s theory of ideas followed Cohen’ theoretical argument in Kant’s Theory of Experience (1871), according to which the a priori “does not merely precede objects”, but “constructs objects” (HCW, I.3, 49). Cohen’s project was to replace the philosophical approaches of sensualism and materialism with the “epistemologically grounded idealism” that he derived from Plato (Cohen 1878, 7) and the modern Platonic tradition. Sensualism was unfit to explain the constructive function of logical and mathematical concepts in science. As Cohen would exemplarily write in The Principles of the Infinitesimal Method (1883) – paraphrasing Descartes – “stars are not given in the sky, but in the reason of astronomy” (HCW 5, 127).18 Cohen made the same epistemological point regarding materialism, which takes the existence of matter as a primitive fact. Cohen’s interpretation of transcendental philosophy was meant as a confutation of this mistaken view (HCW I.3, 270), and can be seen as a late reaction to the “materialism controversy” that raged in German speaking countries since the 1840s, which opposed the supporters of “scientific materialism” and its critics.19 On the whole, the interpretations of Plato and Kant resulted in the postulate of a “dissolution”

17 On

these topics see Pecere 2021. This section and the following are based on this paper reference was to Descartes’ discussion of our sensory and astronomical ideas of the Sun in the Third Meditation (AT VII, 39). 19 For an overview of this “materialism controversy” (Materialismusstreit) see Bayertz et al. (2007) and Beiser (2014), 53–69. 18 The

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of physical matter into the abstract objective correlate of pure thought. Cohen formulated this doctrine in many statements: “There are no things but in thoughts and on the ground of thoughts” (HCW 5, 126). “Mathematics produces the grounds whereby physics can grasp the nature of being. One cannot start with matter; it is a bastard concept” (HCW 5, 31). To be sure, Cohen was aware that these hypotheses required a justification, but he maintained that to solve the “problem of matter” was the “general problem of philosophy” (HCW 5, 35). This philosophical problem was at the same time a historiographical problem. Cohen maintained that “only the historical perspective” can reveal the “logical presupposition of science” (HCW 5, iii). This perspective disclosed the connection between Platonism and the idealist epistemology of modern science. For example, “the role of the category of reality for the concepts of matter and nature” had to be derived not from an abstract investigation of the cognitive faculties, but “from those [philosophers], whose works – deeply interconnected with each other – have led to the discovery of modern science. Galileo, Kepler and Newton, Descartes and Leibniz, with their contemporaries and successors, can teach us how to understand Kant and help us to pursue, in his spirit, the labor of philosophy (HCW 5, iv). These natural philosophers were also self-declared followers of Plato and realized a “rejuvenation” of Plato’s theory of ideas (HCW 5, 26). Hence these founders of modern science confirmed Cohen’s thesis concerning the unifying factor in the history of philosophy, which he restated and generalized in later writings: “the history of idealism in general [ . . . ] is also the history of Platonic idealism. Philosophy is Platonism” (HCW 8, 245). This approach entailed a constraint on the Marburg neo-Kantian historiography of science, whose aprioristic tendency – as I have pointed out in regard to Cassirer – was arguably rooted in a Hegelian background.20 An important example for Cohen was the neo-Kantian Friedrich Lange, whose History of Materialism and Critique of its Present Importance (1866, 1873–52 ) was also meant as a contribution to the materialism controversy in a historico-critical perspective. Lange notably formulated the dialectical argument that a “consequent materialistic view”, if only one investigates the foundations of science, “changes round [ . . . ] into a consequent idealistic view” (Lange 1866, 496). In his Introduction to Lange’s History of Materialism, Cohen spelled out once more his narrative of progress, presenting the path of scientific research as leading “safely and uninterruptedly to idealism” (HCW 5, 92). Cohen’s idea that matter can be “dissolved” (or “resolved”) into forces had a long history in German philosophy, notably in Leibniz’s dynamics, which grounded the corporeal substance of Cartesian philosophy in the activity of fundamental forces, and in Kant’s “dynamical theory of matter” in the Metaphysical Foundations of Natural Science of 1786, which reduces material impenetrability to the action of attractive and repulsive fundamental forces. These doctrines became standard examples of early attempts to solve the problem of matter in the Marburg school, and

20 For

more details see Pecere 2021, 674–675.

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Cohen exemplarily argued that their epistemological objective would be confirmed with the historical development of science.21 This kind of dynamism was reprised and developed by Schelling, Hegel, Adolf Trendelenburg and many others. Lange himself argued that the properties of matter can be reduced to the action of forces. He maintained that “the progressive exactitude of research resolves [auflöst] matter [Stoff ] more and more into forces” and that matter is a “misunderstood residue of analysis” (Lange 1875, 204–205). Other scientific examples discussed by Cohen were Roger Boscovich and Gustav Fechner for their different attempts to combine monads and Newtonian forces in order to explain impenetrability (HCW 5, 135–41). The examination of groundbreaking nineteenth century physics, from the physics of energy to electromagnetism, was supposed to finally provide a scientific confirmation of Cohen’s epistemological hypothesis. Indeed, Cohen argued that Faraday’s theory of electricity led to a true “revolution in the conception of matter and, through the transformation of matter into force, to the victory of idealism” (HCW 5, 71). But Cohen recognized that even Faraday had been just a “forerunner of the new period in natural science” that would only be realized in the electromagnetic theory of mass. In fact these examples, which were also discussed by Cassirer, were not sufficient to prove the “Platonic” direction of modern physics. Kant’s, Boscovich’s and Fechner’s dynamical theories of matter were not examples of mainstream science. On the contrary, corpuscular theories prevailed in modern physics from Galileo and Newton to the nineteenth century. The energetic theory of Ostwald was a generalization of conservation laws that did not entail any reduction of mass to differential equations, as required by Cohen’s epistemology, and the electromagnetic theory of mass was controversial. Cohen himself conceded that he was postulating a yet unavailable physical theory that would be based on a “more elementary [concept], which could serve as the ground of the definitions of mass, force and energy” (HCW 5, 87). History of science had not realized this objective yet. Eventually Cohen, in the third edition of the Introduction to Lange (1914), saluted Einstein’s special relativity as a new achievement in the “history of idealism” because of its abolition of the material ether and the unification of mass and energy. He modified the above quoted passage on the “victory of idealism” (originally referred to Faraday): the victory now consisted in the “transformation of matter into force [and energy]”. In spite of his updating of the previous narrative, Cohen added the triumphant declaration: “the path of research leads with confidence

21 E.g.:

“The materialistic atomism that Leibniz wanted to fight with his monad is rejected by modern mathematical physics” (HCW 5, 134). Cassirer severely criticized Kant’s attempt in Einstein’s Theory of Relativity, where he presents Kant’s metaphysics of corporeal science as a mere “philosophical circumlocution” of the presuppositions of Newton’s physics (ECW 10, 52). This judgement was superficial, for Kant actually took pains to provide an a priori justification of Newton’s claims and in some cases replaced them (e.g. with the rejection of absolute space and the demonstration of gravity as essential property of matter. For full reconstructions of Kant’s dynamical theory of matter see Pecere 2009 and Friedman 2013.

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and without deviations to idealism; at the roots of physical concepts materialism is annihilated” (HCW 5, 91–2, my italics). Paul Natorp followed Cohen’s program by engaging with both the interpretation of Plato and the history of modern science. In Plato’s Theory of Ideas (1903) he developed in considerable detail Cohen’s epistemological interpretation of Plato’s theory, but he pointed out that Plato’s theory of mass (onkos) in the Timaeus, although promising in hindsight, was still very undeveloped (Natorp 1903, 375). In The Logical Foundations of the Exact Sciences (1910, 381–7), he argued that the “dematerialization of matter” could only be realized in modern physics, e.g. in energetics and electromagnetism. Eventually Natorp did not provide an original and detailed account of the connection of Platonism and contemporary science. This task would be faced by Cassirer, who focused on this problem and eventually produced an ingenious interpretation of relativity as a “Platonic” theory.

3.5 Cassirer, Platonism and Relativity Since his first book Cassirer attempted to provide a corroboration of Cohen’s historiographical programme and focused on the “Platonic school” that existed throughout modern science (ECW 1, 66), from Galilei and Kepler to Leibniz. Cassirer sharply pointed out that there was no straightforward connection of Platonism and modern science. First, he argued that in Plato’s dialogues “the possibility of a rigorous and exact science of becoming is straightforwardly denied” and that, in the physics of the Timaeus, “the ultimate explanation of the particular empirical reality is not based on the pure principles of the theory of ideas” (ECW 2, 524). “Only to men of the modern times, only to a Galileo and to a Kepler was possible to be at the same time rigorously Platonic and authentical scientific empiricists” (ECW 2, 527; Cf. 324–325). Second, Cassirer argued that “Platonism” was a sort of ideal standard that modern philosophers and scientists satisfied to different extents. He distinguished different varieties of “Platonism”, concluding that Galileo’s “physical Platonism” was one that “never [ . . . ] had been defended in the history of philosophy and science”, for Galileo “accepted Plato’s hypothetical method but he gave this method a new ontological status; a status which it had never possessed before” (ECW 24, 337, 351). Nevertheless, after his work on Einstein’s relativity, Cassirer formulated the striking hypothesis that “field physics” was the scientific realization of an idea of matter that could be found in Plato’s Timaeus. He presented this thesis in his history of ancient Greek philosophy, published in 1925,22 commenting on the following passage from the Timaeus (49d–50c)23 :

22 Die

Philosophie der Griechen von den Anfängen bis Plato, published in the Lehrbuch der Philosophie, edited by Max Dessoir (Ullstein, Berlin 1925, vol. 1). 23 Engl. transl. by D.J. Zeyl in Plato (2000).

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P. Pecere What we invariably observe becoming different at different times – fire, for instance – to characterize that, i.e., fire, not as ‘this’, but each time as ‘what is such’, and speak of water not as ‘this’, but always as ‘what is such’. And never to speak of anything else as ‘this’, as though it has some stability, of all the things at which we point and use the expressions ‘that’ and ‘this’ and so think we are designating something. For it gets away without abiding the charge of ‘that’ and ‘this’, or any other expression that indicts them of being stable. It is in fact safest not to refer to it by any of these expressions. Rather, ‘what is such’ – coming around like what it was, again and again – that’s the thing to call it in each and every case. So fire– and generally everything that has becoming – it is safest to call ‘what is altogether such’. But that in which they appear to keep coming into being and from which they subsequently pass out of being, that’s the only thing to refer to by means of the expressions ‘that’ and ‘this’. A thing that is some ‘such’ or other, however, – hot or white, say, or any one of the opposites, and all things constituted by these – should be called none of these things [i.e., ‘this’ or ‘that’] [ . . . ] Now the same account, in fact, holds also for that nature which receives all the bodies. We must always refer to it by the same term, for it does not depart from its own character in any way. Not only does it always receive all things, it has never in any way whatever taken on any characteristic similar to any of the things that enter it. Its nature is to be available for anything to make its impression upon, and it is modified, shaped and reshaped by the things that enter it. These are the things that make it appear different at different times. The things that enter and leave it are imitations of those things that always are, imprinted after their likeness in a marvellous way that is hard to describe.

Here is Cassirer’s comment (ECW 16, 448–9): The impression of geometrical forms in the homogeneous, in itself undifferentiated substratum of pure space produces the multiplicity that we design, in the language of our sensory perception, with different sensory qualities taken as a multiplicity of empirical substances. This Platonic physics – bodiless, as it were – wherein all being and all material differences are reduced and dissolved into purely ideal geometrical determinations may appear paradoxical, but then it has to be recalled that not only this physics has been not only reprised in principle by Descartes at the beginning of modern philosophy; its fundamental methodical conception also appears to have found a surprising rebirth in the most recent kind of physics, in that general theory of relativity that ultimately reduces all dynamical determinations to pure metrical determinations [here a footnote was appended: “cf. e.g. Weyl, Space Time Matter”].

This passage is best understood in the context of contemporary discussions on relativity. The connection of general relativity to Descartes’s geometrization of matter was already in Weyl’s text, who argued that “Descartes’ dream of a purely geometrical physics seems to ne attaining fulfilment in a manner which he could certainly have had no presentiment” (1952, 284). In his 1921 book, Cassirer quoted this passage by Weyl and pointed out the “surprising fact” that “very modern physics [ . . . ] seems to be again on the road to Descartes, not indeed in content, but certainly in method” (SR 396, 398). In Philosophy of Mathematics and Natural Science, Weyl would also propose a similar reading of the Timaeus, interpreting both Plato and Descartes with a substantialist language: “spatial extension is the proper substance of bodies” (Weyl 1949, 179. Cf. Timaeus, 48e). Thus Weyl projected the Cartesian res extensa back to Platon’s chôra. Cassirer’s view – as signaled by his remark on “method” versus “content” – was different, for the mathematization here was taken

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as a sign of a reduction of sensory objects and pictures to pure relations that was precisely opposed to the “substantialist” talk in science.24 In order to make Cassirer’s position stand out in this context, it is important to mention Émile Meyerson’s interpretation of relativity. In The Relativistic Deduction (1925), Meyerson also presented a connection between relativity and mathematical Platonism.25 He pointed out that Weyl had aptly presented the geometrical character of general relativity as “a sort of amalgam of the theories of Newton and Pythagoras [ . . . ] Since, however, Weyl is stressing here the geometrical nature of this panmathematicism, it would perhaps have been appropriate to add Plato’s name to that of Pythagoras, for, as we know, it was he who gave Pythagoreanism a geometrical form” (Meyerson 1925, 152). Meyerson’s epistemological view was that sensations “result from a persistent and unique reality” which lies “behind these appearances”. He maintained that this was originally a Platonic teaching.26 Meyerson also observed that the history of physics appears as “the constant realization of the Idea, in the Platonic sense of the term. Despite incessant contradictions inflicted on it by reality, the Idea tends to impose itself upon our conception of reality – to force reality to enter into the mold of the Same” (196). As we have seen Cassirer – who was well-acquainted with Meyerson’s work since Substance and Function – was not far from the latter’s teleological view of history, but he disagreed with Meyerson’s search for a Parmenidean “identity” in nature and the conflation of the latter view with the epistemological teaching of Plato. Cohen’s and Natorp’s researches had shown that Plato, in his late dialogues, had rather started the resolution of sensory things into relations between multiple forms. This was the background of Cassirer’s interpretation of the Platonic passage as reducing bodies to “purely ideal geometrical determinations”. Cassirer’s interpretation and reconsideration of the Timeus represents, I submit, the culmination of the historical-theoretical research on Platonism and relativity in the Marburg school. Drawing on the Marburg interpretation of Plato, Cassirer detected an actual Platonic motive in some of the philosophically most prominent scientists and philosophers of his time, like Einstein, Weyl, Eddington, Meyerson and Whitehead. The merits and limits of this interpretation can be best ascertained

24 In

Substance and Function, Cassirer argued that Descartes’ famous account of the piece of wax in the Second Meditation was insufficient: after reducing the sensory object to extension, the latter had to be “reduced to a pure phenomenon of simple and individual centers of force” (ECW 6, 177–178). 25 The premises of this connection were already set out in previous works. In Explanation in the Sciences (1921), Meyerson connected Plato’s Timaeus to Descartes’ geometrical explanation of phenomena (Meyerson 1921/1991, 97–98, 216–218), and in the same context he mentioned Einstein’s relativity as a possible corroboration of Kantian idealism (ivi, 409). 26 “Plato already realized this, showing that when different observers conceive differently the size and shape of one and the same thing, it is still possible, by means of number and measurement, to form a unique concept that explains this diversity” (Meyerson 1925/1985, 19). The reference was to Resp 602c–603a.

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with the help of a further passage from the Phenomenology of Knowledge (ECW 13, 540)27 : The reality that we designate as a ‘field’ is no longer a complex of physical things, but an expression for an aggregate of physical relations. When from these relations we single out certain elements, when we consider certain of its positions by themselves, it never means that we can actually separate them in intuition and disclose them as isolated intuitive structures. Each of these elements is conditioned by the whole to which it belongs; in fact it is first defined through this whole. It is no longer possible to separate an individual part, a substantial particle, from the field and follow the movement of these parts for a certain time. Here, then, the method of defining a physical ‘object’ by a mode of ‘indication’, a tode ti, however subtle, is precluded from the very first. This form of demonstration fails, and must be replaced by a far more complex form of physical deduction. In the ether of modern physics – as Eddington declared on occasion – we can no longer set our finger on a definite place and maintain that this or that one of its parts was in this place a few seconds ago.28

This quote echoes the above quoted passage from the Timaeus: we recognize the critique of sensory perception and the resolution of properties of physical objects into relations belonging to a physico-mathematical whole – the field of contemporary physics –, which realizes the epistemological idea of Plato’s chôra. Still, Cassirer’s subtle interpretation did not solve the problem of Marburg’s program. Indeed, as we have seen with respect to Weyl, there was still a gap between Cassirer’s epistemological ideal and the historical reality of physics.

3.6 Conclusions The case study that I have examined shows how the controversy over materialism influenced the neo-Kantian epistemology of the Marburg school and prepared the interpretation of Einstein’s theory or relativity and the program of a unified field theory in the 1920s. The critique of materialism was a driving force in late nineteenth century and early twentieth century German epistemology, which had deep cultural and social motivations besides the purely epistemological side (Köhnke 1991). To be sure, although the connection of anti-materialism and defence of spiritual freedom was still significant in Cassirer and Weyl, their epistemological arguments on matter as a physical concept stood on their own ground and contributed to a lively debate among philosophers and scientists. The limit of Cassirer’s perspective (shared with Cohen) was rather the tendency to take for granted the success of an unfinished program for the sake of its epistemological meaning.

27 Engl. 28 This

tr. by R. Manheim (Cassirer 1957, 465). expression would also be used by Weyl concerning the identity of the electron (1949, 171).

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Einstein was foreign to the anti-materialist background of Cassirer’s views, but he could sympathize with some of their epistemological points. In a letter to Cassirer of 1937, Einstein praised the philosophers’s presentation of contemporary physics in Determinism and Indeterminism in the Modern Physics (1937). Einstein focused his comment on a passage on Leibniz’s rejection of atomism as inconsistent with a “representation according to continuous functions”, arguing that this view was “ingenious” and that “in due time” Leibniz would be proved to be “right” (in Cassirer 1996–2021, XVIII, 158–9). This might have sounded like a confirmation of the “Platonic” view of Marburg, which considered Leibniz as the modern champion of idealism and the connecting figure between Plato and Kant, preparing the ground for the “pure field physics”. However, Einstein was thinking of the discrete representation of matter in the new quantum mechanics and of his unaccomplished project of replacing this theory with a new system of differential equations. Moreover, Einstein’s project of a replacing quantum mechanics was directed by a realist epistemology. He might agree that discrete matter had to “dissolved” into continuous fields “in due time”, but was very far from celebrating the “victory of idealism”.

References Bayertz, Kurt, Miriam Gerhard, and Walter Jaeschke, eds. 2007. Weltanschauung, Philosophie und Naturwissenschaft im 19. Jahrhundert, vol. 1. Der Materialismus-Streit. Hamburg: Meiner. Beiser, Frederick. 2014. After Hegel: German Philosophy, 1840–1900. Princeton: Princeton University Press. Cassirer, Ernst. 1923. Substance and Function; Einstein’s Theory of Relativity. Chicago: Open Court. ———. 1957. The Philosophy of Symbolic Forms. Vol. 3. The Phenomenology of Knowledge. New Haven/London: Yale University Press. ———. 1998–2009 (=ECW). Gesammelte Werke, hrsg. von B. Recki. Hamburg: Meiner. ———. 1995–2021. Nachgelassene Manuskripte und Texte, 19 vols. Hamburg: Meiner. Cohen, Hermann. 1878. Platons Ideenlehre und die Mathematik. Marburg: Pfeil. ———. 1987–2002 [= CW]. Werke, ed. Hermann-Cohen-Archiv unter Leitung von Helmut Holzhey. Hildesheim: Olms. Descartes, René. 1897–1913 [=AT]. Oeuvres de Descartes, ed. C. Adam and P. Tannery. Paris: Leopold Cerf. Eddington, Arthur. 1920. Space Time and Gravitation. Cambridge: Cambridge University Press. Einstein, Albert. 1917. Über die spezielle und die allgemeine Relativitätstheorie (Gemeinverständlich). Braunschweig: Vieweg. English Translated by R.W. Lawson, R.W. 1920. Relativity: The Special and the General Theory. New York: Holt & Co. ———. 1922. Remarks. Bulletin de la Société Française de Philosophie 22(3): 91–113. Reprinted in La Pensée 210: 12–29 (1980). ———. 1953. The Meaning of Relativity. Princeton: Princeton University Press. Einstein, Albert, and Michele Besso. 1972. Correspondance 1903–1955. Paris: Hermann. Einstein, Albert, and Infeld Leopold. 1967. The Evolution of Physics, 1st ed., 1938. New York: Touchstone. Ferrari, Massimo. 2006. Il Kant degli scienziati. Immagini della filosofia kantiana nel tardo Ottocento tedesco. Rivista di Storia della Filosofia 61 (4): 183–201.

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Friedman, Michael. 2013. Kant’s Construction of Nature. Oxford: Oxford University Press. Galilei, Galileo. 1968. Opere. Firenze: Barbera. (Reprinted). Giovanelli, Marco. 2003. August Stadler interprete di Kant. Guida. Gower, Barry. 2000. Cassirer, Schlick and ‘Structural Realism’: The Philosophy of the Exact Sciences in the Background to Early Logical Empiricism. British Journal for the History of Philosophy 8 (1): 71–106. Harman, Peter M. 1982. Energy, Force and Matter. The Conceptual Development of NineteenthCentury Physics. Cambridge: Cambridge University Press. Howard, Don A., and Marco Giovanelli. 2019. Einstein’s Philosophy of Science. In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta, Fall 2019 Edition. https://plato.stanford.edu/ archives/fall2019/entries/einstein-philscience/. Jammer, Max. 1997. Concepts of Mass in Classical and Modern Physics. Mineola: Dover. Kant, Immanuel. 1900. Kants gesammelte Schriften, hrsg. von der Königlich Preußischen Akademie der Wissenschaften. Berlin et al.: De Gruyter. Kim, Alan. 2015. Neo-Kantian Ideas of History. In New Approaches to Neo-Kantianism, ed. N. De Warren and A. Staiti, 39–58. Cambridge: Cambridge University Press. Köhnke, Klaus Christian. 1991. The Rise of Neo-Kantianism. Cambridge: Cambridge University Press. Ladyman, James. 2020. Structural Realism. In The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta, Winter 2020 Edition. https://plato.stanford.edu/archives/win2020/entries/ structural-realism/. Ladyman, James, and Don Ross. 2010. Every Thing Must Go. Metaphysics Naturalized. Oxford: Oxford University Press. Lange, Friedrich Albert. 1866. Geschichte des Materialismus und Kritik seiner Bedeutung in der Gegenwart, 2nd ed., 2 vols, 1873 (vol. I) and 1875 (vol. II). Iserlohn: Baedeker. Lange, Marc. 2002. An Introduction to the Philosophy of Physics. Locality, Fields, Energy and Mass. London: Blackwell. Maxwell, Gower. 1970. Structural Realism and the Meaning of Theoretical Terms. In Analyses of Theories and Methods of Physics and Psychology, Minnesota Studies in the Philosophy of Science, ed. M. Radner and S. Winokur, vol. 4, 181–189. Minneapolis: Minnesota University Press. Meyerson, Émile. 1921. De l’explication dans les sciences. Paris: Payot. English Translation Kluwer, Dordrecht, 1991. ———. 1925. La dÉduction relativiste. Paris: Payot. English Translation Kluwer, Dordrecht, 1985. Natorp, Paul. 19212 [19031 ]. Platos Ideenlehre. Leipzig: Meiner. English Translation by V. Politis and J. Connolly. Sankt Augustin: Academia Verlag, 2004. ———. 1910. Die logischen Grundlagen der exakten Wissenschaften. Leipzig/Berlin: Teubner. Pais, Abraham. 1982. «Subtle is the Lord . . . » The Science and the Life of Albert Einstein. Oxford: Oxford University Press. Pecere, Paolo. 2009. La filosofia della natura in Kant. Bari: Edizioni di Pagina. ———. 2021. History of Physics and the Platonic Legacy: A Problem in Marburg NeoKantianism. British Journal for the History of Philosophy 29 (4): 671–693. https://doi.org/ 10.1080/09608788.2021.1881443. Plato. 2000. Timaeus. T´ιμαιoσ. Indianapolis/Cambridge: Hackett. Poincaré, Henri. 1908. La science et l’hypothèse. Paris: Flammarion. Ryckman, Thomas. 2005. The Reign of Relativity. Philosophy in Physics 1915–1925. Oxford: Oxford University Press. Schilpp, Paul A., ed. 1949 (19693 ). Albert Einstein: Philosopher-Scientist. La Salle Ill: Open Court. Scholz, Erhard. 2006. The Changing Concept of Matter in Hermann Weyl’s Thought 1918–1930. In Interactions, ed. V.F. Hendricks et al., 281–306. Dordrecht: Springer. Weyl, Hermann. 1949. Philosophy of Mathematics and Natural Science. Princeton: Princeton University Press.

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———. 1952. Raum Zeit Materie [19181 ]. English Translation of the 4th Edition by H. L. Brose. Mineola: Dover. Whitehead, Alfred North. 1919. An Enquiry Concerning the Principles of Natural Knowledge. Cambridge, MA: Cambridge University Press. Worrall, John. 1989. Structural Realism: The Best of Both Worlds? Dialectica 43: 99–124. Reprinted in Papineau, D., ed. The Philosophy of Science, 139–165. Oxford: Oxford University Press.

Chapter 4

Cassirer and Klein on the Geometrical Foundations of Relativistic Physics Francesca Biagioli

Abstract Several studies have emphasized the limits of invariance-based approaches such as Klein’s and Cassirer’s when it comes to account for the shift from the spacetimes of classical mechanics and of special relativity to those of general relativity. Not only is it much more complicated to find such invariants in the case of general relativity, but even if local invariants in Weyl’s fashion are admitted, Cassirer’s attempt at a further generalization of his approach to the spacetime structure of general relativity seems to obscure the fact that the determination of this structure requires a completely different approach that is derived from Riemann rather than Klein. This paper aims to reconsider these issues by drawing attention to the combination of subtractive and additive strategies (more in line with Riemann’s) in Klein’s own considerations about the determination of the structure of spacetime from 1897 to 1910. I will point out that Cassirer relied on Klein’s argument in some central passages from Substance and Function (1910), and elaborated further on his combined approach in Einstein’s Theory of Relativity (1921), also taking into account the application of Riemannian geometry in general relativity. My suggestion is that an appreciation of Cassirer’s continuing commitment to a variety of geometrical traditions may shed light on his particular understanding of a priori elements of knowledge and avoid some of the classical objections to the idea of a relativization of the a priori. Keywords Spacetime geometry · Scientific change · Relativized a priori · Ernst Cassirer · Felix Klein

F. Biagioli () University of Turin, Turin, Italy © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Russo Krauss, L. Laino (eds.), Philosophers and Einstein’s Relativity, Boston Studies in the Philosophy and History of Science 342, https://doi.org/10.1007/978-3-031-36498-3_4

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4.1 Introduction The German mathematician Felix Klein and the neo-Kantian philosopher Ernst Cassirer are known to have been two of the main proponents of an invariancebased approach, according to which the content of a given geometry is what remains invariant under a transformation group, and an analogous consideration helps to determine the geometry of spacetime in physics. Klein first outlined his grouptheoretical reconstruction of geometry from a purely mathematical point of view in his Erlangen Program of 1872. He engaged in the debate over the geometrical foundations of physics beginning at the end of the 1890s, and made important contributions to the mathematical framework of relativistic physics, in particular his paper of 1911 on the geometrical foundations of the Lorentz group and his intervention in the discussion over the first cosmological models of general relativity in 1918. But it was especially Cassirer who emphasized the general scope of an invariance-based approach along Klein’s line in his first major epistemological work, Substance and Function (1910), as well as in his later works, including the third volume of the Philosophy of Symbolic Forms (1929), the fourth volume of The Problem of Knowledge (1950 [1940]), and “On the Concept of Group” (1944). Not only did Cassirer point to Klein’s projective model of non-Euclidean geometry and to the group-theoretical view as the mathematical basis for a general account of the different hypotheses concerning the form of space in physics, but he characterized his epistemology as a universal invariant theory of experience searching for ideal common elements of historical scientific theories in analogy with the mathematical investigation of the relative invariants of a transformation group. With his work of 1921 on the epistemological implications of general relativity, Cassirer sided with eminent interpreters of Einstein’s theory such as Hermann Weyl and Arthur Eddington. All these authors sought to extend the invariancebased approach to infinitesimal geometry and identified the generalized topological invariants of general relativity as a prioris (in the relativized sense of constitutive elements of a particular theory), thereby proposing a Kantian alternative to the interpretation of the new physics put forward in logical positivism.1 However, several studies have emphasized the limits of invariance-based approaches when it comes to account for the shift from classical mechanics and of special relativity to general relativity (from now on, GR).2 Not only is it much more complicated to find such invariants in the case of GR, but even if local invariants in Weyl’s fashion are admitted, Cassirer’s attempt at a further generalization of his approach to the spacetime structure of GR seems to obscure the fact that the determination of this structure requires a completely different approach that is derived from Riemann rather than Klein. To Einstein’s contemporaries the use of

1 On

Cassirer’s role in this line of interpretation, see esp. Ryckman (1991, 2005, Ch. 2). a reconstruction of Einstein’s philosophy of science, see esp. Howard in Howard and Stachel (1989). For the analysis of the limits the subtractive strategy associated with invariance-based approaches, I will rely especially on Norton (1999). 2 For

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Riemann’s geometry appeared to be one of the most revolutionary ideas of the new physics. As Cartan put it: General relativity threw into physics and philosophy the antagonism that existed between two principle directors of geometry, Riemann and Klein. The space-times of classical mechanics and of special relativity are of the type of Klein, those of general relativity are of the type of Riemann. (Cartan 1927. Engl. Transl. in Norton 1999 p. 128)

Norton spells out Cartan’s remark by suggesting that invariance-based approaches along Klein’s line are associated with a subtractive strategy, which consists in starting from the consideration of the mathematical structure determined by a group and reading off the invariants while abstracting from all the properties that depend on the mathematical formalism. By contrast, Riemann started with the concept of an n-fold extended manifold whose intrinsic properties are just continuity and dimensionality, and added the geometric notion of length by means of a quadratic differential form. This allowed him to extend the Gaussian method to deal with surfaces of variable curvature. When applied to physics, Riemannian approach suggested an additive strategy, which consists in locating the metrical properties of the spacetime manifold in a quadratic differential form of the line element. Norton’s suggestion is that Einstein’s theory of gravitation shows a combination of these two strategies: While following Riemann in the determination of metrical properties, he relied on Klein, insofar as he deprived his coordinate system of all but topological properties. In Norton’s reconstruction, the latter subtractive strategy underlies Einstein’s remarks of 1916 on the physical content of his theory, in particular the expansion of the covariant group. This tension opened the door to the debate over the correct understanding of Einstein’s relativity principles that informed also the more recent discussion on the relativization of the a priori (besides the already mentioned works by Ryckman, see esp. Friedman 1999, 2002; Stump 2015). While Cassirer is usually acknowledged as one of the first to have paved the way towards this relativization, his account of a priori elements in terms of relative invariants has been charged with offering a purely regulative account of a priori knowledge that is completely detached from the way in which scientific concepts actually change. One of the main objections against his interpretation of general relativity in particular is to have downplayed the physical significance of the line element, among other things because of his continued reliance on the geometrical tradition that goes back to Klein. This paper aims to reconsider the fact that there is no clear-cut distinction between these two strategies in nineteenth-century geometry, but a combination of both can be traced back to Klein’s considerations on the determination of the structure of spacetime in a series of contributions from 1897 to 1911. I will point out that both aspects of Klein’s approach played an important role in Cassirer’s account of the concept of space in Substance and Function (1910), and shed further light on his appreciation of the role of Riemannian geometry in Einstein’s Theory of Relativity (1921). Cassirer’s continuing commitment to a variety of geometrical traditions will suggest a possible line of response to some of the classical objections against the relativization of the a priori, in particular the irrelevance of universal invariants.

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4.2 Felix Klein’s Argument from Projective Geometry In Norton’s reconstruction, Klein’s subtractive approach was modeled on projective geometry and the geometries of metrical space of zero or constant curvature. In 1871 he presented a model of non-Euclidean geometry based on the analytic treatment of projective geometry by means of the algebraic theory of invariants. Klein used a projective metric first studied by Arthur Cayley to derive the three principal cases of geometries of metrical space of constant curvature, that is parabolic, elliptic, and hyperbolic (corresponding to a manifold of null, more or less than zero, respectively).3 He gave a fully general presentation of his approach in his “Comparative Review of Recent Research in Geometry”, now best known as the “Erlangen Program”. Klein’s work first appeared as a pamphlet distributed during his inaugural address as newly appointed professor at the University of Erlangen in 1872. Using a concept first introduced by Evariste Galois, Klein characterized the task of geometry as that of, given a manifold and a group of transformations of the same; to develop the theory of invariants relating to that group. “Transformation” indicates a one-to-one mapping of space onto itself. Translations and rotations are typical examples of transformations of ordinary geometry, which leave invariant parallelism, lengths, and the measures of angles. By contrast, projections or collineations leave invariant only such relations as, of points, to lie on the same line, etc. Transformations form a group iff: (i) the product of any two transformations of the group also belongs to the group; (ii) for every transformation of the group, there exists in the group an inverse transformation.4 These formulations suggest that each group is of equal importance with every other. In order to clarify the relations of one group to another, Klein introduced the following principle of transfer. Suppose that a manifold A has been investigated with reference to a group B. If, by any transformation whatever, A is converted into a second manifold A , the group B of transformations, which transformed A into itself, will become a group B , whose transformations are performed upon A . The principle states that every property of a configuration contained in A obtained by means of  to be obtained by the the group B corresponds to a property of a configuration in A

3 Cayley in his “Sixth Memoir upon Quantics” (1859) had limited himself to show that his projective system included (Euclidean) metrical. Klein was the first to derive the non-Euclidean cases (i.e., elliptic and hyperbolic geometries). 4 Klein specified the second condition in the 1893 version of the paper. In the first version of 1872, he adopted Jordan’s (1870, p. 22) definition, which referred to finite groups of permutations. In that case, the first of the conditions above, along with the existence of a neutral element, is sufficient for the set to form a group. Afterwards, Sophus Lie drew Klein’s attention to the fact that the existence of an inverse operation is required in the case of infinite groups. It is noteworthy that both Galois and Jordan dealt with groups of permutations. Klein showed that the same conditions for permutations to form a group generally apply to groups of operations. In this sense, his Erlangen Program can be considered a fundamental step in the development of the abstract concept of group (Wussing 1984).

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group B . According to Klein, this principle shed light on the arbitrariness of the choice of a particular space element or the number of dimensions: As there is nothing at all determined at the outset about the number of arbitrary parameters upon which these configurations shall depend, the number of dimensions of our line, plane, space, etc., may be anything we like, according to our choice of the element. But as long as we base our geometrical investigation upon the same group of transformations, the geometrical content remains unchanged. (Klein 1893, p. 73. translation slightly modified)

The latter passage is one of the clearest expressions of the structural conception of geometrical figures that Klein compared to the numerical representation of projective space. The projective model of non-Euclidean was the first example of how the same principle of transfer can be used to prove the equivalence of different geometries by identifying the underlying group of transformations. From this point of view, Euclidean and non-Euclidean geometries are classified as subgroups of the projective group. Klein’s ideas remained largely unknown until the first translations of the Erlangen Program into Italian (1890), French (1891), and English (1893), and Klein’s revised version of 1893. Klein (1893, p. 63, note) reported that he was motivated to promulgate his earlier project by Lie’s Theory of Transformation Groups, which appeared in three volumes in 1888, 1890, and 1893. Lie’s study of continuous transformation groups provided essential requirements for the implementation of the project of a unified treatment of geometry from the standpoint of group theory.5 Subsequently, as a leading figure in the mathematical culture of his time, Klein reported in his collected mathematical papers to have been able to resume an early project to spell out the physical as well as philosophical implications of his classification of geometry.6 He wrote: In the same period [the 1890s] I was able to cultivate my interests in mechanics and mathematical physics, as I had been intending to do since the beginning of my mathematical studies. The first physical investigations in the theory of relativity emerged a few years later, and rapidly attracted general attention. I suddenly recognized that my classification of 1872 included even these investigations and provided the simplest way to clarify the newest physical (or even philosophical) ideas from a mathematical viewpoint.7 (Klein 1921, p. 413)

Klein addressed the epistemological question of geometry in a number of occasion in the 1990s, from his concluding remark to his paper of 1990 “On

5 On

Klein’s relationship to Lie, see Hawkins (1984) and Rowe (1989). first job, before turning to purely mathematical research, was to set up and carry out demonstrations accompanying Julius Plücker’s lectures in experimental physics. Klein worked as Julius Plücker’s assistant at the University of Bonn from 1866 to 1868. When Plücker died, in 1868, Alfred Clebsch appointed Klein at Göttingen to complete the posthumous edition of Plücker’s work on line geometry. This position enabled Klein to move into the circle of algebraic geometers inspired by Clebsch. On the development of Klein’s thought in the early stages of Klein’s career, see Rowe (1992) and Gray (2008). 7 All translations from Klein’s texts are my own, unless otherwise indicated. 6 Klein’s

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Non-Euclidean Geometry” – a mathematical paper providing a new solution to the problem of defining the class of surfaces in elliptic, hyperbolic, and parabolic space that are locally isometric to the Euclidean plane, to his 1897 review of Lie’s work when this was awarded the first Lobachevsky prize.8 The 1900s saw a further development of Klein’s view in connection with the emergence of Einstein’s special theory of relativity. Hermann Minkowski in a celebrated lecture from 1908 titled “Space and Time” adopted the group-theoretical approach to characterize the spacetime geometry of special relativity. Klein pointed to Minkowski’s spacetime formulation as a new application of his geometric methods in his public lecture of 1910, “On the Geometrical Foundations of the Lorentz Group”, which was published with the same title in Physikalische Zeitschrift. The following passage from this paper contains the clearest statement of a substantial connection between the group-theoretical approach to geometry and relativistic physics: What the modern physicist call “relativity theory” is the theory of invariants of a four dimensional spacetime region, x, y, z, t (the Minkowski “world”) with respect to a particular group of collineations, namely the “Lorentz group”; or more generally, and turned round the other way: If one wants to make a point of it, it would be all right to replace the phrase “theory of invariants relative to a group of transformations” with the worlds “relativity theory with respect to a group”. (Klein 1911, p. 21)

The reconstruction of Klein’s EP as “subtractive” offers a way to check whether an algebraic expression has geometrical significance; however, it does not account for the “constructive” side of his approach. Following Schiemer’s (2020) suggestion, Klein’s understanding of geometric content as determined by the group can be spelled out in terms of an abstraction principle stating that the types of two figures in a manifold are identical if they are congruent relative to the transformation group. Two figures F 1 , F2 of a manifold M are congruent relative the group G, if there exists a transformation f in G such that f(F1 ) = F2 . The principle states that: T ype (F1 ) = T ype (F2 ) , iff F1 is G − congruent to F2

.

This suggests an interpretation of Klein’s approach as an in rebus structuralism, according to which mathematical theories describe abstract structures depending for their existence on their instantiation in concrete mathematical systems. A clear example of this way of proceeding is offered by Klein’s projective model of non-Euclidean geometry. Klein presented his model in a series of papers published in 1871, 1873, 1874 and repeatedly emphasized its epistemological significance in the 1890s and 1900s. In this connection, he also emphasized that the advantage of starting with the analytic representation of space depends on the fact that a segment of the projective line can be uniquely correlated with the series of real numbers. To put it in modern terminology, there is a local isomorphism between a unidimensional projective form and subsets of the real numbers. It is noteworthy that Klein in his Collected Mathematical Works from 1921 still referred to his 1873

8 Klein’s

review appeared in Mathematische Annalen in 1898.

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paper on non-Euclidean geometry for the characterization of “invariant theory” in connection with what he called “the projective way” of treating a manifold (Klein 1921, p. 122). My suggestion is that the numerical representation of projective space opened the door to a consideration of space as a more general structure than the Euclidean case – which in Klein’s classification corresponds to parabolic geometry. Klein explored the possibility of specifying the same structure in different ways. This possibility bears on Klein’s view of measurement, because it suggests that different hypotheses have to be taken into account when it comes to the structure of physical space at the infinitesimal level as well as in the infinitely large. His 1898 paper gives evidence of the fact that Klein was aware of the distinction between the subtractive strategy that is derived from his classification of geometries and the idea of a step-wise procedure to formulate the various hypotheses concerning the curvature of space. While it is true that Klein’s conception of geometry was modeled on results concerning the more restricted class of spaces of constant curvature, he did rely on this model to explore both a subtractive strategy and an additive strategy more in line with Riemann’s. Klein described his distinction as determined by the starting point that is being taken in the characterization of space. The first possibility is to start from the presupposition of the numerical representation of space as an n-dimensional manifold and a definition of the underlying group. Klein pointed out that this is the way taken, for instance, in metrical geometry, when one presupposes the free mobility of rigid bodies (i.e., the axioms of congruence) (Klein 1898, p. 588). This way of proceeding clearly illustrate the characteristics of the subtractive strategy in Norton’s sense. Klein started with the most general characterization of space using a numerical structure, and showed how the geometry under consideration can be determined independently of the arbitrary coordinate assignment by specifying the underlying group and the relative axiomatic definitions. The various hypotheses concerning the curvature of space can be then narrowed down by criteria of conformity to observation and intuitive expectations as well as conventions. Klein wrote: When it comes to take into consideration the topologically different forms of space for the determination of the geometry of actual space, we are faced not so much with an arbitrary but with an inner consequence. Our empirical measurement has also an upper limit, which is given by the dimensions of the objects accessible to us or to our observation. What do we know about spatial relations in the infinitely large? To begin with, absolutely nothing. Therefore, we rely on the postulates that we formulate. I consider all of the different topological forms of space equally compatible with experience. The fact that in our theoretical considerations we put first some of these forms of space (i.e., the original types, that is, the properly parabolic, hyperbolic, and elliptic geometries) and finally select parabolic geometry (i.e., the usual Euclidean geometry), depends solely on the principle of economy. (Klein 1898, p. 595)

However, Klein also went on to show how a complementary (additive) strategy can be gained from the way in which numbers are being “introduced” in projective geometry, that is assigned to a geometrical constructions in a univocal way. Klein described the correlation of the points of a given curve with the numbers of a unidimensional continuum as a process of idealization from concrete and inexact

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intuition to the formulation of exact geometrical statements. Within the reach of empirical intuition, each point actually corresponds to a particular number, and vice versa. One postulates that the correspondence between points and numbers should take place not only with empirical precision, but in an absolute sense, for the rational numbers, subsequently for the reals and, finally, for any number. Klein used the fact that the metrical scale of Euclidean geometry is a special case of the projective scale to highlight how the non-Euclidean hypotheses may be introduced starting from the consideration of a finite region of space. This would consist in taking into account all the hypotheses concerning the infinitesimal and the infinitely large that according to Klein’s model would be compatible with the Euclidean ones at a local level. Klein emphasized this point also in the second volume of his Elementary Mathematics from a Higher Standpoint, which was written in the same period as the paper on the geometrical foundation of the Lorentz group. Talking about the introduction of projective coordinates in the way that we have just sketched, Klein wrote: We especially appreciate the fact that, in order to introduce the coordinates, we do not need the whole projective space and we can restrict every consideration to a finite region of space. Such a restriction has a principled interest, because it takes into account that, in the application of geometry to the space of empirical intuition, we are not warranted to make statements about spatial elements at any distance; in particular the discussion about whether parallels exist or not does not make any sense from the very beginning. (Klein 1909, p. 154)

Cassirer referred to the same procedure in Substance and Function (1910) to show how a generalized a priori of space can be derived from the resolution of spatial concepts into pure serial concepts. Cassirer wrote: As in the case of number we start from an original unit from which, by a certain generating relation, the totality of the members is evolved in fixed order, so here we first postulate a plurality of points and a certain relation of position between them, and in this beginning a principle is discovered from the various applications of which issue the totality of possible spatial constructions. In this connection, projective geometry has with justice been said to be the universal “a priori” science of space, which is to be placed besides arithmetic in deductive rigor and purity. Space is here deduced merely in its most general form as the ‘possibility of coexistence’ in general, while no decision is made concerning its special axiomatic structure, in particular concerning the validity of the axiom of parallels. Rather it can be shown that by the addition of special completing conditions, the general projective determination, that is here evolved, can be successively related to the different theories of parallels and thus carried into the special “parabolic,” “elliptic” or “hyperbolic” determinations. (Cassirer 1910/1923, p. 88)

In the wake of Cohen’s reading of Kant, Cassirer’s philosophy of the concept of function shifts the focus from the notion of a priori elements (forms of intuition, concepts, and judgments) to that of a priori synthesis. Cassirer articulated the theory of experience into a multi-level hierarchy of conceptual syntheses rather than following the Kantian articulation into categories of the understanding and forms of intuition. At the basic level, exemplified by the number series, the particular mathematical objects are determined by an ordering function. A further step is given by the coordination of different ordered series, as in the example of projective coordinates. Finally, Cassirer outlined how these structural techniques culminated

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with the study of mathematical structures and classes of structures in the current sense of abstract concepts that can have a variety of instantiations. A typical example of these kinds of concepts was offered by Klein’s treatment of geometry as the study of the constructions that remain unvaried under transformation groups. In the light of such a treatment, the notion of Euclidean three-dimensional space was included in a more comprehensive theory of the forms of space that according to mathematicians such as Klein and Poincaré provided the most basic presuppositions for the possibility of measurement in physical space. The development of the theory of transformation groups allowed a generalization of the Kantian notion of the “form of all spatial intuition” from the Euclidean group to the wider group of collineations, as first suggested by Helmholtz (1878).9 Subsequently, Cassirer recognized that GR called into question even these generalized accounts of the form of space, and that his own argument for the applicability of mathematical syntheses to the empirical manifold as formulated in 1910 deserved to be reconsidered in some important respects. Before turning to Cassirer’s interpretation, it is important to point out that, nevertheless, the core idea of Cassirer’s relativization of the a priori was already in place in his account mathematical concept formation in functional terms, when considered in its different steps, and not only from the viewpoint of the resulting scientific representation of the phenomena in terms of structures. Cassirer’s fundamental syntheses are intrinsically relational and dynamical, insofar as they make possible the conceptual determination of something that is initially taken as indeterminate.

4.3 Cassirer and the Change of Geometry in Relativistic Physics Cassirer offered a comprehensive discussion of the philosophical implications of GR in Einstein’s Theory of Relativity (1921). It is not the aim of this paper to reconstruct Cassirer’s interpretation or defend it against all the objections that have been raised against it, beginning with Schlick’s (1921).10 The purpose of this section is to point out that, insofar as a combination of subtractive and additive strategies was already in place in his previous argument in the wake of Klein, he was able to offer a further articulation of it so as to account for the epistemological impact of GR. In particular, I will point out – against what is sometimes taken to be the standard reconstruction of Cassirer’s view in the contemporary discussion on the relativized a priori – that Cassirer in this connection appreciated the additive strategy that is made available

9 Helmholtz’s role in the philosophical background of GR has been emphasised in different ways in the literature. See esp. Coffa (1991), Friedman (1999), Ryckman (2005), DiSalle (2008). On Helmholtz’s influence on Cassirer’s earlier generalizing strategy before 1921, see Biagioli (2016). 10 For a thorough discussion of Cassirer’s stance within the philosophical debate on GR, see Ferrari (1991), Ryckman (1991, 2005).

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by the use of Riemannian geometry, and recognized the general (non-Euclidean) form of the line element as a contentful constitutive element of the new physics. Cassirer started by pointing out that geometrical conventionalism had received a fundamental confirmation in a twofold way. The first concerns Poincaré’s (1902) argument that physical theories have to do with metrical relations and not with the concepts of space and time considered in themselves. In Cassirer’s interpretation, this was also a consequence of Einstein’s insight into the methodological nature of the preconditions of measurement. As Einstein famously put it, GR took from space the “last remnant of physical objectivity” (Einstein 1916, p. 117). Cassirer rephrased Einstein’s thought by saying that with GR the problem of determining what space is in itself has become meaningless, and has been replaced by the methodological question concerning the use of different systems of geometrical propositions “in the interpretation of the phenomena of nature and their dependencies according to law” (Cassirer 1921/1923, p. 439). At the epistemological level, according to Cassirer, there is only the question: “Whether there can be established an exact relation and coordination between the symbols of non-Euclidean geometry and the empirical manifold of spatio-temporal events” (p. 433). Cassirer’s suggestion is for the epistemologist to acknowledge the fact that GR has answered such a question affirmatively. In Cassirer’s neo-Kantian epistemology, the recognition of this fact imposed a further generalization of the “a priori” of space to the function of spatiality expressed in the general concept of the linear element ds. Cassirer departed from Kant by acknowledging that such a function belongs to the conceptual rather than to the intuitive conditions of knowledge. Cassirer had already granted the categorial status of space in the 1900s with respect to Helmholtz’s attempt at a generalization of the Kantian form of outer intuition to the three classical cases of manifolds of constant curvature. Cassirer’s further revision of 1921 allowed him to reaffirm the grounding function of spatiality in the constitution of physical objectivity. The second aspect of Poincaré’s argument that can be confirmed according to Cassirer is that Euclidean geometry retains a privileged status in GR, in the Euclidean form of the line element for the infinitesimally small. Cassirer referred to the fact that gμν in the general form can be disregarded at the infinitesimal level and in all cases where the laws of special relativity retain their validity in GR. These are the cases where one can assume a homogenous gravitational field containing an inertial system, and so transform back the form of the linear element into the Euclidean form on physical grounds. According to Cassirer, this way of proceeding confirms that, all other things being equal, GR continues to rely on the simpler hypothesis of a space everywhere homogeneous. Unlike the previous theories though, it also requires the more complicated hypothesis of Riemannian geometry to account for the relations of measurement holding in general in relativistic physics. On the one hand, the functionalist account of objectivity received a new confirmation from the fact that the mathematical and conceptual tools for the new theory came right from what was considered to be one of the most highly abstract developments of nineteenth-century geometry. Not only Poincaré, but none of the scientists who had taken into consideration the applicability of non-Euclidean

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geometry in physics before GR, had envisioned the application of Riemannian geometry in physics. On the other hand, the impulse for the search of more complex hypotheses to master the relativistic problem of measurement came from the physics. Cassirer wrote: That the elements, to which we must ascribe, methodologically, a certain “simplicity” must be adequate for the interpretation of the laws of nature, cannot be demanded a priori. But even so, thought does not simply give itself over passively to the mere material of experience, but it develops out of itself new and more complex forms to satisfy the demands of the empirical manifold. (Cassirer 1921/1923, p. 438) When we look back over our earlier considerations, this view must undergo a peculiar and paradoxical reversal. Pure Euclidean space stands, as is now seen, not closer to the demands of empirical and physical knowledge than the non-Euclidean manifolds but rather more removed. For precisely because it represents the logically simplest form of spatial construction it is not wholly adequate to the complexity of content and the material determinateness of the empirical. Its fundamental property of homogeneity, its axiom of the equivalence in principle of all points, now marks it as an abstract space; for, in the concrete and empirical manifold, there never is such uniformity, but rather thorough-going differentiation reigns in it. (Cassirer 1921/1923, p. 443)

What justifies this reversal in Cassirer’s view is the possibility of a more universal coordination of geometry and physical reality, which accounts for the laws of the previous theories as still valid under determinate conditions. Cassirer took the idea of a process of successive approximation here as a confirmation of the possibility of the a priori synthesis of knowledge. As Cassirer put it, the absolute differential calculus was developed by Gauss, Riemann and Christoffel as an immanent “progress of pure mathematical speculation”; but now that it gained a surprising application in Einstein’s theory of gravitation, it also “serves directly as the form into which the laws of nature are poured” (Cassirer 1921/1923, p. 400). That the relation between mathematical and physical syntheses in Cassirer’s understanding is an entangled one emerges clearly also from his remarks about the nonlinear direction of the progress of knowledge, where “the way upward and the way downward” (i.e., abstraction and determination) appear at some points as “one and the same” (Cassirer 1921, p. 443). Cassirer emphasized in continuity with the main insight of his philosophy of the concept of function, that the physicist seems to gain a “richer” picture or reality, corresponding to higher standards of precision of empirical measurements, in as much as she turns to the highest mathematical abstractions for the formulation of laws. His interpretation of GR complicated this picture further, insofar as he acknowledged the possibility of reversals in the order of syntheses for a more adequate conceptualization of the empirical manifold. In this regard it is worth noting that Cassirer did not refer to the calculus of tensors in its final form (which the absolute differential calculus assumed only in conjunction with the establishment of GR), but to the developments of the fundamental ideas of Gauss’s theory of surfaces and of Riemann’s geometry throughout the nineteenth century. In Cassirer’s dynamical understanding of the system of knowledge, mathematics and physics co-evolve in various ways, with a constant tendency to provide a comprehensive construction of external reality.

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A complementary (subtractive) strategy can be retraced in Cassirer’s attempt to show the advantage of Einstein’s generally covariant theory of gravitation over the previous theories, and to link it with his functional account of objectivity. Cassirer argued that GR had reached a new standpoint in the functionalization of physical objectivity with the requirement of the general covariance of physical laws. The fact that there cannot be global inertial frames in Einstein’s gravitational theory indicates that the general relativity of motion based on the principle of equivalence must be formulated with full spacetime coordinate generality. Cassirer contended that the general covariance of physical laws in the case of GR works not only as a formal requirement on mathematical representation, but as “a principle which the understanding uses hypothetically as a norm of investigation in the interpretation of experience” (Cassirer 1921, p. 415). What this norm entails, in Cassirer’s view, amounts to assuming the unity of nature and the exact determination of it. By assimilating under arbitrary transformations of the coordinates all measurement results obtainable in particular reference systems, GR eliminated the anthropomorphic element that was inherent in the previous theories’ reference to the idea of a background spacetime, and allowed for a univocal characterisation of spatiotemporal events in their relation to other structures in the field. Relying on Ryckman’s analysis (2005, pp. 42–46), one can say that general covariance, in Cassirer’s understanding, offered another example of a principle that has both a regulative meaning (as the methodological norm stating that the laws of nature find their only natural expression in generally covariant equations) and a constitutive one, insofar as physical objects are identified with what remains invariant under arbitrary transformations of the coordinates. Cassirer held that a formal choice about coordination (e.g., a choice of gauge) turns out to be necessary to constituting a certain set of observables. A suitable example that Ryckman (2018) considers is the way in which Cassirer (1936) considers the transformation of the concept of “physical state” as definable only in abstract symbolic terms in quantum mechanics. The case of the general covariance is arguably more complicated. Einstein himself seems to have struggled to show the physical significance of general covariance of the field equations of GR in the years immediately following the presentation of his theory. Kretschmann famously objected to him that general covariance, when understood as mere coordinate generality, is nothing more than a formal constraint on the theory’s mathematical form, and has nothing to do with the principle of general relativity or with the theory of gravitation. The received view that Einstein gave up his earlier efforts to show the physical significance of general covariance has been challenged by a line of research initiated by Stachel, emphasizing that the fundamental purpose of general covariance from Einstein’s remarks of 2016 to his attempts to clarify his view after Kretschmann’s objection, was to avoid any principled separation of the metrical and the underlying topological structure of spacetime (see e.g., Howard and Stachel 1989). Ryckman (2005) uses insights from this research to reconsider Cassirer’s appreciation of the epistemological innovation of the theory of relativity. Cassirer’s and others’ attempts to identify general covariance as a contingent but constitutive a priori principle, however, can hardly be taken to reflect Einstein’s philosophical

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views. While recognizing that Ryckman shows that Cassirer, along with Weyl and Eddington, should be read as having offered a self-consciously Kantian alternative to logical empiricism, Howard (2005) points out that neither alternatives seem to have appealed to Einstein.11 Without addressing all of these issues here, the following section will draw on the combination of additive and subtractive strategies that we have seen at work in Cassirer’s account of the geometry of GR to contrast his view with the reception of it in the contemporary discussion on the relativization of the Kantian a priori.

4.4 Some Concluding Remarks on Cassirer and the Relativized A Priori Friedman’s (2005) suggestion is to reconsider Cassirer’s (and Reichenbach’s) arguments for dealing with the post-Kuhnian problem whether it is possible to escape the conclusion that all knowledge is local, given the incommensurability of succeeding scientific paradigms such as the shift from Newtonian mechanics to relativistic physics. Friedman’s answer is affirmative. However, it requires him to rephrase these neo-Kantian arguments as follows. Cassirer’s contribution to the discussion of the post-Kuhnian problem according to Friedman is to offer an ideal reconstruction of the connection between past and present theories in terms of an abstract mathematical relation of approximate inclusion. Such a relation is illustrated in mathematical cases such as the derivation of Euclidean geometry as a limiting case of a projective determination of measure also including a variety of non-Euclidean cases. We have seen that Klein’s projective model of non-Euclidean geometry offered a characteristic example of what Cassirer called coordination between different ordered series in (1910). Friedman’s reconstruction in set-theoretic terms emphasizes the abstract relation of inclusion of the special cases into the model.12 By the same token, one can say that GR contains both special relativity and Newtonian physics, insofar as the laws of these theories can be derived as approximate limiting cases of Einstein’s later theory of gravitation. Cassirer’s view in Friedman’s reconstruction would amount to assume that the constitutive a priori principles of succeeding theories tend to converge in a similar way towards the purely regulative ideal of a maximally general structure. The problem with the argument thus reconstructed is that it does not account for discontinuities in the very formulation of the constitutive principles. In the case of GR, there is not only a generalization of Euclidean metric to the form of the line element in a semi-Riemannian structure, but metrical geometry shifts from

11 For a reconstruction of Einstein’s philosophy of science and further references, see esp. Howard and Giovanelli (2019). 12 Cf. Biagioli (2020) for a reconstruction of Cassirer’s argument elucidating the dynamical aspect of Cassirer’s notion of coordination.

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playing the role of a constitutive element to being determined depending on the distribution of mass and energy. With this example in mind, Friedman proposes to reconsider Reichenbach’s (1920) account of discontinuities in terms of revisions of the principles coordinating the abstract mathematical structures to physical reality. A related problem, according to Friedman, is that both Cassirer’s and Reichenbach’s accounts of theory change are backward directed, and only become possible after the change has taken place. But they do not offer what Friedman would consider to be a “rational” explanation of the change itself. In order to address this problem, Friedman’s suggestion is to draw on the Habermasian notion of a “communicative rationality” integrating the ‘instrumental’ rationality of the exact sciences when the scientific community decides to abandon a research program in favor of another.13 A philosophical account of these dynamics, in Friedman’s view, would have to look for continuities across theory change at the meta-scientific level of the transformation of concepts. Contrary to what Friedman suggests, however, the main purpose of the transcendental method of the Marburg School was to provide an account of how scientific objects are constituted.14 Apart from the interpretative issues raised by Friedman’s reconstruction though, my suggestion is that his objections miss the main point of Cassirer’s argument from the geometry of GR. This example shows that conceptual change was the main target of Cassirer’s account of the constitution of scientific objectivity at different but interconnected levels. Once understood in relativized terms, the relations between these levels must be distinguished from backward-directed inclusion, and there is no need to look at a nonscientific notion of rationality for a forward-oriented account of scientific change. This is not to say that Cassirer held the same view as Reichenbach. Cassirer’s argument is that what changed in the transition from special to general relativity was our realizing that a specific metrical structure of spacetime could no longer be taken as a priori and that, instead, what must be regarded as a priori in GR is a weaker topological structure, one compatible with different varieties of metrical structure. In Reichenbach’s view, GR shows that the way in which such a change in what is taken to be a priori can occur is not by subsumption but by wholesale replacement. With wholesale replacement it is harder to make the case for continuity and, therefore, rationality in theory change. This is why Friedman must look elsewhere for conceptual resources to account for the dynamics of reason. On the opposite side, Stump maintains that relative a priori cognitions in Reichenbach’s sense are better understood as fallible and theory-specific constitutive elements, which in Stump’s account include conventions as well as empirical factors 13 See

Friedman (2001, Ch. 3) for an account of Habermas’s distinction and for the idea to integrate the relativized a priori with the Habermasian notion of communicative rationality for a comprehensive account of the dynamics of reason throughout the history of the sciences. 14 Several studies have shown that Cassirer, and his Marburg teachers before him, did attribute a constitutive function to a priori structures of scientific experience while introducing a dynamical dimension at some level of scientific conceptualization (see esp. Ferrari 2012; Giovanelli 2016; Biagioli 2020).

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that are being presupposed in the practice of science. Given the variety of such elements and their local validity, Stump’s proposal is to drop all claim to universality and necessity that is characteristic of the Kantian a priori. Therefore, Stump’s suggestion is to abandon the notion of “a priori” altogether, while exploring further the potential of the notion of “constitution of objectivity” (Stump 2015, p. 3). Stump dismisses Cassirer’s strategy, insofar as he takes it to imply that the a priori principles of theoretical physics can be relativized while a core of purely mathematical notions remain a priori in an “absolute” sense. Even though a two-tier account of a priori cognition can accommodate the transformation of the concept of space in modern physics, Stump points out that what scientists actually presuppose are concepts that change from one historical period to another, and not what these concepts “really are in some Platonic heaven” (Stump 2015, p. 94). The above objection, which clearly echoes Schlick’s and Friedman’s, only holds if Cassirer’s view is taken to separate mathematically fixed forms common to all scientific experience and relativized principles that are specific to different scientific domains and subject to change from one theory to another. However, there is no such separation in Cassirer’s view, as it emerges from the fact that his system allows for reversals in the order of the mathematical concepts that are supposed to be more or less suited to establish a univocal coordination of physical experience. Not only is it not the case that the relativization of the a priori is necessarily backward-directed, but Cassirer’s account offers interesting insights for a discussion that has focused mainly on the problem to elucidate the continuities (or the discontinuities) of one theory with respect to another, rather than on the internal relations of the various constitutive elements among themselves. We have seen that mathematics thereby is taken to fix the space of the logical possibilities that are open to all theories, whereas the principles of physics can change from one theory to another. My suggestion is that Cassirer emphasized that there can be also a co-evolution of mathematics and physics that is not always taken into account in the contemporary discussion on theory change, but that would deserves closer attention in cases such as the relation of non-Euclidean geometry and GR. Acknowledgments This research has received funding from the “Rita Levi Montalcini” program granted by the Italian Ministry of University and Research (MUR). I would like to thank the editors of this collection and an anonymous referee for their helpful comments on a previous draft of this paper. I also wish to remark that the current paper is my own work, and no one else is responsible for any mistakes in it.

References Biagioli, F. 2016. Space, Number, and Geometry from Helmholtz to Cassirer. Cham: Springer. ———. 2020. Ernst Cassirer’s Transcendental Account of Mathematical Reasoning. Studies in History and Philosophy of Science Part A 79 (2020): 30–40. Cassirer, E. 1910. Substanzbegriff und Funktionsbegriff: Untersuchungen über die Grundfragen der Erkenntniskritik. Berlin: B. Cassirer. English edition in Cassirer, E. 1923. Substance and

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Function and Einstein’s Theory of Relativity. Trans. M. C. Swabey, and W. C. Swabey. Chicago: Open Court, 1–346. ———. 1921. Zur Einstein’schen Relativitätstheorie: Erkenntnistheoretische Betrachtungen, 347– 465. Berlin: B. Cassirer. English edition in Cassirer E. 1923. Substance and Function and Einstein’s Theory of Relativity. Trans. M. C. Swabey, and W. C. Swabey. 347–465, Chicago: Open Court. ———. 1936. Determinismus und Indeterminismus in der modernen Physik: Historische und systematische Studien zum Kausalproblem. Göteborg: Göteborgs Högskolas Årsskrift 42. English edition: Cassirer, Ernst. 1956. Determinism and Indeterminism in Modern Physics: Historical and Systematic Studies of the Problem of Causality. New Haven: Yale Univ. Press. ———. 1944. The Concept of Group and the Theory of Perception. Philosophy and Phenomenological Research 5: 1–36. ———. 1950. The Problem of Knowledge: Philosophy, Science, and History since Hegel. Trans. W. H. Woglam, and C. W. Hendel. New Haven: Yale Univ. Press. Coffa, A. J. 1991. The Semantic Tradition from Kant to Carnap: To the Vienna Station. Cambridge: Cambridge University Press. DiSalle, R. 2008. Understanding Space-Time: The Philosophical Development of Physics from Newton to Einstein. Cambridge: Cambridge University Press. Einstein, A. 1916. Die Grundlagen der allgemeinen Relativitätstheorie. Annalen der Physik 49: 769–822. English edition: Einstein, A. 1923. The Foundation of the General Theory of Relativity. Trans. G. B. Jeffrey and W. Perrett. In The Principle of Relativity, ed. H.A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, 111–164. London: Methuen. Ferrari, M. 1991. Cassirer, Schlick e l’interpretazione ‘kantiana’ della teoria della relatività. Rivista di filosofia 82: 243–278. ———. 2012. Between Cassirer and Kuhn: Some Remarks on Friedman’s Relativized A Priori. Studies in History and Philosophy of Science Part A 43: 18–26. Friedman, M. 1999. Reconsidering Logical Positivism. Cambridge: Cambridge University Press. ———. 2001. Dynamics of Reason: The 1999 Kant Lectures at Stanford University. Stanford: CSLI Publications. ———. 2002. Geometry as a branch of physics: Background and context for Einstein’s ‘Geometry and experience’. In Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathematics, ed. D. B. Malament, 193–229. Chicago: Open Court. ———. 2005. Ernst Cassirer and Contemporary Philosophy of Science. Angelaki Journal of Theoretical Humanities 10 (1): 119–128. Giovanelli, M. 2016. Zwei Bedeutungen des Apriori‘. Hermann Cohens Unterscheidung zwischen metaphysischem und transzendentalem a priori und die Vorgeschichte des relativierten a priori. In Philosophie und Wissenschaft bei Hermann Cohen/Philosophy and Science in Hermann Cohen, ed. C. Damböck, 177–204. Cham: Springer. Gray, J. J. 2008. Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton: Princeton University Press. Hawkins, T. 1984. The Erlanger Program of Felix Klein: Reflections on Its Place in the History of Mathematics. Historia Mathematica 11: 442–470. Helmholtz, H. v. 1878. Die Tatsachen in der Wahrnehmung. In Schriften zur Erkenntnistheorie, ed. P. Hertz and M. Schlick, 109–152. Berlin: Springer, 1921. English edition as Epistemological writings (Trans: Lowe, M. F., ed. R. S. Cohen and Y. Elkana). Dordrecht: Reidel, 1977. Howard, D. 2005. “No Crude Surfeit”: Ryckman’s Reign of Relativity. Pacific APA , March 2005. Howard, D., and M. Giovanelli. 2019. Einstein’s Philosophy of Science. Stanford Encyclopedia of Philosophy, ed. E.N. Zalta and U. Nodelman. Accessed on 22 July 2023: https:// plato.stanford.edu/entries/einstein-philscience/ Howard, D., and J. Stachel, eds. 1989. Einstein and the History of General Relativity. Boston: Birkhäuser. Jordan, C. 1870. Traité des substitutions et des équations algébriques. Paris: Gauthier-Villars. Klein, F. 1871. Über die sogenannte Nicht-Euklidische Geometrie. Mathematische Annalen 4: 573–625.

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Chapter 5

Natorp, Cassirer and the Influence of Relativity Theory on Neo-Kantian Philosophy Luigi Laino

Abstract In this paper, I will survey the “received view” of the interpretation of relativity theory in Natorp and Cassirer. Neo-Kantian and non-neo-Kantian scholars (such as Hentschel or Ferrari) usually distinguish Natorp’s reading from Cassirer’s by virtue of “immunising” and “revising” strategies. “Immunisation” consists of a strict defence of Kantian philosophy, while “revision” pertains to the modification of Kantianism depending on relativity theory. In this respect, I will suggest some arguments that will put things in perspective. In particular, I will show that Natorp’s interpretation is justified considering the state of physical research in 1910. By the same token, I will highlight where Cassirer leverages immunising strategies. However, I will demonstrate that, in contrast to Natorp, the influence of general relativity (GR) is pivotal to Cassirer and it does have an impact on his whole epistemology (and philosophy), implying a highly radical reform of pure intuition in light of general covariance. I will also add that Cassirer may have a bearing on Einstein as to the possibility of reconsidering his former censure of Kantian philosophy. Keywords Natorp · Cassirer · Einstein · Relativity · Covariance

Abbreviations If possible, Einstein’s works and correspondence have been cited according to the Princeton edition of the Collected Papers. Other sources are included in the reference list. Cassirer’s unpublished correspondence is available both in the DVDROM attached to the Nachlass and on the Internet (https://agora.sub.uni-hamburg. de/subcass/digbib/ssearch).

L. Laino () University of Naples Federico II, Naples, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Russo Krauss, L. Laino (eds.), Philosophers and Einstein’s Relativity, Boston Studies in the Philosophy and History of Science 342, https://doi.org/10.1007/978-3-031-36498-3_5

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CP The Collected Papers of Albert Einstein (John Stachel et al., Eds.). Princeton, NJ: Princeton University Press, 1987–present. CBW Ausgewählter wissenschaftlicher Briefwechsel. In NMW––Cassirers Nachgelassene Manuskripte und Werke (J. M. Krois, Ed.), 18. Hamburg: Meiner, 2009. CGW Gesammelte Werke von Ernst Cassirer (B. Recki et al., Eds.), Hamburger Ausgabe. Hamburg: Meiner, 1998–2009. Bd. 1–26. KrV Kritik der reinen Vernunft (1966). Stuttgart: Reklam; Critique of Pure Reason (P. Guyer & A. W. Wood, Transl.). Cambridge: Cambridge University Press, 1998. Original work published in 1781 and 1787.

5.1 Introduction: The Received View of the Neo-Kantian Interpretations of Relativity Theory In the history and philosophy of physics, neo-Kantian and Einsteinian scholars paid and pay much attention to Cassirer’s interpretation of relativity theory, but are less interested in Natorp’s early reading of it. This hinges on historical reasons: Natorp published Die logischen Grundlagen der exakten Wissenschaften in 1910 and could deal prevalently with special relativity (SR) and the Minkowskian spacetime formalism (Howard 1994: 50). Therefore, his interpretation would be outdated, or even flawed in comparison with Cassirer’s remarks, which revolve instead around GR. The first aim of my paper is to reconsider Natorp’s position and to suggest that it was justified for his time. This will automatically shed new light on what I call the “received view”, that is, the idea that Natorp belongs to those “radical Kantians” (Reichenbach 1922: 23) who conceived of relativity as a prosecution of transcendental philosophy.1 In fact it is generally argued that Natorp maintained the distinction between transcendental space and time and their empiric counterparts, which is at odds with Einstein’s program, but I will show that this was acceptable in 1910. Nevertheless, I will not ignore that Cassirer’s interpretation appears to be more in line with Einstein’s vision on relativity. Thus, the second objective of my paper is to evaluate how far Cassirer pushes the readjustment of Kantianism in order to cope with GR’s demands and constraints. Since there are already valuable contributions on the topic, I will try to bring something new to it, especially by drawing attention to Cassirer’s relationship with Natorp. We will see that Cassirer

1 Truth be told, Hentschel distinguished three paths concerning “immunising strategies” and four “revising” paths––Hentschel (1990: 199–239); see also Ferrari (1996: 111–146). For the purpose of this paper, we can simply assume that to immunise Kant means to defend the existence of transcendental space and time at all costs, while to revise him implies the proclivity to accept significant revisions to balance the transcendental and the empirical part of knowledge.

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Table 5.1 The list provided by Hentschel (1990)

exploits both immunising and revising arguments, a combination which makes his revision peculiar. As far as the methodology of this paper, I will leverage Hentschel’s classic tools in relativistic epistemology, which are summed up below (Table 5.1). Yet, I find it helpful to add the following to my exposition. In line with what I explained above, I propose evaluating interpretations in accordance with the historical development of physical theory. It follows that it would be unfair to judge Natorp’s reading on the basis of GR: there was not an established theory of it when Natorp wrote his book. It is conversely striking that in 1910 Natorp showed rather good comprehension of Minkowki spacetime. Of course, that does not mean to underestimate criteria such as flexibility or breadth. That is, if an epistemological interpretation cannot survive revolutions in physics, it is not as powerful as one which can. Furthermore, it might be reasonable to reflect on what I propose to call “impacting criteria”. One should deal with, on the one hand, (12) the development of a reliable model of scientific progress (which, however, partially contradicts (4)); and on the other, (13) the impact of relativity theory upon the whole system.2 These elements should do justice to the genuinely philosophical ambitions of the neoKantians (Table 5.2). Finally, I come to the structure of the paper. In the second section, I will deal with Natorp’s interpretation of relativity theory, while in the third, I will focus on Cassirer’s revision of transcendental philosophy in the light of GR, and compare and contrast his vision with Natorp’s. I will explain the rationale that I have followed in formatting sub-sections at the beginning of each section.

2 Krois has already highlighted, with regard to relativity, Cassirer’s tendency to favour the spontaneous reaction of different research fields (2009: XXIX).

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Table 5.2 The list supplied with impacting criteria

5.2 Natorp Though relativity theory impacts the mathematical definition of space and time, I cannot address Natorp’s sophisticated reading (Natorp 1910: 276, but see also: 266; 274–275) as this would require a lengthy explanation. For this reason, I will directly concentrate on his interpretation of relativity. Nevertheless, I will also deal with Natorp’s assessment of meta-geometry, which I find may well show what sort of revising evolutions are possible when departing from a strict immunising narrative.

5.2.1 Natorp’s Early Interpretation of Relativity Theory To begin with, it is clear that Natorp’s main reference is Minkowski rather than Einstein. In his book, Natorp only mentions Zur Elektrodynamik bewegter Körper (1905). This is not surprising since Minkowski’s classic essay Raum und Zeit (1909) contains the following explosive statements for a Kantian or a neo-Kantian philosopher. First, (i) one should consider that space and time are conflated into a unique manifold that Minkowski himself calls “the world” (Minkowski 1909: 79), subtly suggesting, I think, that one would be in a position to reject Kant’s distinction between “appearances” and “things in themselves”. Second, (ii) Minkowski claims that Euclidean, to wit, three-dimensional geometry is but a chapter in fourdimensional physics. As far as (i) is concerned, it seems to cast a shadow on Natorp’s general conviction that one can maintain Kant’s absolute space and time. But even worse, (ii) deals with Natorp’s analysis of the concept of univocal coordination between

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axioms and experience and appears to hinder Natorp’s claims about the judgement of existence.3 Thus, one should ask whether or not it is so. Natorp’s general strategy is to show that relativity is but a “konsequente Durchführung” of critical philosophy. Needless to say, one achieves such immunization after having sided with a particular vision of Kant. Indeed, Natorp assumes as “absolute” space and time their purely geometrical representation as distinct from the empirical one affected by relativity: We recognise in Minkowski’s relativity principle only the forceful implementation of the difference between the pure, absolute and mathematical determination in space and time (Zeit- und Raumbestimmung)––as it was already deployed by Newton, just as held and more precisely conceived by Kant––from the empirical and physical one, which can be solely relative. Without reason, one believed to challenge that difference starting from new intuitions; on the contrary, it is undoubtedly confirmed by the principle, though it is further sharpened and stringently formed in its implementation. (Natorp 1910: 399)

Natorp reasons this way to reject the relativist stance of phenomenalists such as Mach. If one comprehended the effects of length contractions and time dilations as real, one would indeed not know how to distinguish mathematical from physical elements. On the contrary, the usage of c in the Lorentz transformation allows that. C is not an empiric fact; rather, it is the condition through which we determine experience up to the limit of the speed of light in a vacuum. In Natorp’s parlance, it is a “Grenzbegriff ”. Accordingly, the velocity of light enables the formulation of the equations and works as their condition of possibility––in Cassirer’s words, it will become a “system fact” instead of a “perception fact” (1920/1921: 86). In this respect, what one would now call the Lorentz factor––to wit, .γ =  1 ––plays 2

1− v2 c

the same role as differentials and derivatives in calculus: it is the “absolute” with regard to which we conceive of any experience to be possible. That is: according to SR, we base objective representations of natural phenomena upon the mathematical fixation of the light propagation, while beyond it there is a lack of determinism in the definition of relations between events in space and time.4 Clearly, this is but the causal structure, viz. the so-called light-cone representation of SR (Eddington 1921: 53). As already stated, it is therefore Minkowski spacetime which is at play. Minkowski underlined that a projection in space and time is still possible from the “world”, viz. from the 4d manifold (Minkowski 1909: 82; Ihmig 2001: 169). 3 In Natorp, the “Existenzurteil” has the following sense. In the first place, it leaves nothing undetermined (Natorp 1900: 370, 1910: 274–276). In the second place, it does not coincide with the single object since all one knows about existence is that it is asserted (loc. cit.: 301). It follows that one must not conflate the “univocality” (Einzigkeit) of the object into its “singularity” (Einzelheit). Existence is then defined this way: “Existence is but the expression whereupon all thought strives for. Indeed, thinking means to determine, and existence means the last determination within which nothing remains undetermined” (Natorp 1910: 305). 4 In Natorp’s words: “The velocity of light appears throughout as an ultimate factor that similarly enters as a condition all of our time- and space-measurements. There is no possibility that it will show not to be constant, as long as it is not given to us a superluminal measure of time- and space-determination” (1910: 395).

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Space and time are, indeed, distinct vector-components that we combine through the definition of proper time, a scalar that is independent from coordinates  and follows the equation: .dτ = 1c c2 dt 2 − dx 2 − dy 2 − dz2 (1)––the integral τ = τ will represent the time of a worldline. From this point forward, it suffices to apply first and second derivatives to get the vector of motion and that of acceleration, which are respectively time- and space-like. Minkowski, however, has shown that the introduction of  · t = s, where c is considered equal to 1, makes space- and time-coordinates homogenous (1909: 86). For his part, Natorp assumes that relativity postulates the relative changes of both time- and space-parameters, not their conflation: The time-parameter does not become homogeneous with space-coordinates; its relationship to each of these remains essentially different from the mutual relationship of the coordinates. The “union” about which Minkowski speaks only rests on the circumstance that the four parameters, mutually and within given functional constraints (in bestimmter Abhängigkeit voneinander), revealed the same relativisation. (Natorp 1910: 398)

In practice, one has to distinguish the analytic injection of time into the manifold from the determination of physical reality in space and time;5 in a manner of speaking, one has to filter Descartes’ triumph through Kant (Natorp 1910: 397– 398). As to the aprioricity of geometry, there are no criteria that prevent us from conceiving of a non-Euclidean coordination between axioms and experience. Yet, Lorentz transformations are orthogonal and despite the perceptive extravagancies foreseen by the equations, they are represented within a semi-Euclidean and continuous group (M4 ). As Einstein will say, we can consider Minkowski’s world as a 3d Euclidean space plus one imaginary time-coordinate (Einstein 1920: 37; Rovelli 2021: 26–27). Since the geometrical meaning of i is but a 90◦ rotation of the axes in the Cartesian plane, it might not be a stretch to contend that preliminary concepts such as direction, counter-direction and rotations are sufficient to generate the switch of coordinates and work as its condition of possibility (Eddington 1921: 53 and ff.). It follows that one begins with a Euclidean-based mathematical construction which turns into a non-Euclidean and consistent physical diagram through the adjustment of specific concepts and operations. Although this appears to be a misunderstanding (see Torretti 1996: 88), for Natorp’s argument undermines the import of non-classical transformations (Hentschel 1990: 220), Natorp’s immunisation is not monotonic. One encounters, in particular, two topics of interest, i.e.: (iii) a developmentalist standpoint on the nature of scientific concepts, and (iv) the question of invariance. As far as (iv) is concerned, suffice it to recall that Natorp believes the factum of science to be a fieri, so the condition of possibility of the constancy of the speed of light is removable (Natorp 1910: 402). This explains why Natorp finally rejects Minkowski’s idea of a pre-

5 Natorp

asserts that it is variation in magnitudes that creates time and not the opposite (Natorp 1910: 200–208; 331); this also makes time appear in the equations of mechanics only as numerical series (loc. cit.: 282).

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established harmony between mathematics and physics (Minkowski 1909: 88) as a dogmatic one. Accordingly, one finds evidence of a developmentalist epistemology: Pure mathematics does not relate with physical experience so as to make possible a direct agreement. [ . . . ] Of course, the mathematics of natural science––that is: the premises that make mathematics applicable to physics in a given stage of its development––must exactly agree with the grounding premises of this physics; however, there is no further instance than the mathematical natural science itself through which we establish or may establish what harmony is at play. (Natorp 1910: 401)

However, the ambiguity regarding the distinction between “Eindeutigkeit” and “Einzigkeit” as to the coordination of geometry and physics deserves more detailed discussion. Although Howard (1992) thinks they are synonymous, and to a large extent it is so, I propose translating “Eindeutigkeit” with “unambiguousness” and “Einzigkeit” with “univocality” to stress that “Einzigkeit” compels us to make more rigid decisions about the mathematical construction of physical reality.6 Indeed, the demand of “univocality” somewhat mirrors a principled defence of the systems to which Kantian philosophy owes its fortune. This is the case when Natorp emphasises that Newtonian equations are saved when one can hold c as mathematically equivalent to ∞. Instead of highlighting that Newtonian laws are special cases of expanded laws of relativity, he claims that they are valid as soon as we replace Gc with G∞ (Natorp 1910: 397–398). Other excerpts seem to point in a different direction by acknowledging the import of Lorentz transformations (loc. cit.: 403-4), but it is clear that such a position impinges on the definition of invariance as the pinnacle of the revision of Kant’s philosophy. These major issues with Natorp’s interpretation reverberate in his approach to gravitation. I am referring to a passage where Natorp extends the consideration of the limiting process c → ∞ to Newton’s universal law. In a nutshell, he ignores the introduction of the weak equivalence principle (1907) and does not mention the mental experiment of the rotating disk, which suggested the idea that the metric of a rotating frame of reference––in motion with respect to a Galilean one––is variable and not strictly Euclidean. But it would be too much to condemn Natorp on this point: indeed, claims on the rotating disk, introduced by Born and Ehrenfest, are well-established only around 1912 (Ryckman 2017: 203; 207 and ff.). Actually, Natorp may once again have Minkowski’s paper in mind. Minkowski wrote an equation that describes the relation between the worldlines of two masspoints m and m1 . Though he only considered the special condition according  to  which the acceleration vector of m is null, one would have (2): .−mm1 γ − x˙c1 R. And then (3): .c Rτ − Rx = r12 , .Ry = cy2¨r , Rz = 0. If one takes into account magnitudes that do not involve the velocity of light, then τ → 1 and one gets approximations to Kepler’s laws (Minkowski 1909: 88). Natorp thus creatively reads that as if relativity did not object to the validity of classical laws; rather, it would mould their limitedness more sharply.

6 The

demand of “univocal determination” has been already introduced in Natorp (1900: 389).

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To a certain extent, Cassirer will expand on Natorp’s claims, and try to carefully revisit the dependence of classical from generally invariant laws. To understand how this becomes possible, it is worthwhile addressing Natorp’s interpretation of metageometry. This will facilitate us in bridging the gap between Natorp’s conservative reading on Minkowski spacetime and Cassirer’s full acknowledgment of general covariance.

5.2.2 Natorp’s Assessment of Meta-Geometry If one is to fully understand Natorp’s reading of meta-geometry, one should tackle Natorp’s assessment of geometry. Needless to say, this is beyond the scope of this paper. However, my hypothesis is that at least a cursory enquiry into Natorp’s sources may help us follow Cassirer’s line of thought on the neo-Kantian revision of GR. This would suffice to point out the difference between immunisation and revision. At the time when Natorp was writing, the word “meta-geometry” or “metamathematics” was pejorative and carried the reference to both empiristic and metaphysical claims about non-Euclidean geometry (Biagioli 2016: 67, foot. n. 15). In particular, Natorp believes that the simultaneous assumption of the “manyworlds” stance as to the logical possibility of any given geometry and the empirical reality of a singular system is controversial. Indeed, on the one hand, it is a mistake to conceive of a priori elements as real; on the other, infiniteness about logical possibility would entail that all representations are equally real (Natorp 1910: 336– 337). Moreover, although Natorp is aware that Kant’s standpoint is not ultimate (loc. cit.: 318), he is convinced that one should not give up three Kantian ideas: (1) space and time are not empiric concepts; (2) they are not discursive either; (3) they do not stem from “comparative universality”, which is a simply empiric procedure to gain concepts through abstraction from the particulars. That being said, Natorp compares his standpoint to Kant’s and endorses three major results of Kant’s enquiry: (K1 ) Euclidean space is neither a necessity of thought nor (K2 ) a fact of experience, for in both cases its truth would depend on a given empiric test, which is impossible. Quite the opposite, space (K3 ) is a necessity of the thought of experience, that is: it enables the objective foundation of experience for it makes the latter possible. When this happens, one shifts from logical to real possibility or from pure thought to existence. Therefore, there is no need for assuming either the many-worlds hypothesis or the empiricity of geometry; suffice it to hold to the univocality of coordination. But what does it mean to univocally coordinate axioms with experience? Let us start from the evidence that one already has in 1910 to conceive of the empiric consistency of non-Euclidean geometry. Usually, proofs in this field were connected to the nature of human intellect. Poincaré explains the deformation of the Euclidean continuum through physical hypotheses––for instance, the warming of rigid bars in the periphery of such a space––, which he judges may fit the intelligence

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of beings other than humans or humans educated in a different way7 (Poincaré 1908: 52 and ff.). At the same time, Natorp could not accept proofs that would mistake mathematical for psychological intuition such as von Helmholtz’s. But more in general, if one understands the generalisation of non-Euclidean geometries as stemming from a Euclidean background, the elevation of the concept of invariance to a new status does not eventuate in the refutation of Euclidean geometry. On the contrary, in this way one will prove its mediating power between pure intellect (the realm of all possible geometries) and experience (given by measurements but filtered through Euclidean constructions). In short, the principle of a free coordination of axioms with experience requires limitation. The latter is due to the mathematical construction one needs as soon as one thinks of intermediate functions that instantiate, as it were, the intellectual syntheses in the phenomena. In particular, it seems that one has to work on principles that would play a constitutive or “constructive” function in showing how one applies mathematical structures to experiences (Natorp 1905: 8).8 Among them, one finds the definition of rigid bodies, the rectilinear propagation of inertial motions and that of the speed of light in a vacuum, which are, in 1910, the basic tools to draw a Minkowski diagram. Thus, Natorp can still infer that “these assumptions presuppose in turn a given geometry, that is, Euclidean geometry in its usual fashion; how could those measurements decide the validity of a given geometry which is their premise?” (Natorp 1910: 314). In this way, it is Natorp’s conviction that he has sided with Poincaré’s statement that one cannot prove the truth of geometry through experience (Poincaré 1908: 93–95). This notwithstanding, Natorp’s immunisation has side effects. The idea of a transcendental and Euclidean construction of non-Euclidean geometries casts a shadow on the “developmentalist” model of the history of physics, according to which the progress of knowledge consists of the application of invariance criteria to increasingly wide ranges of phenomena. Indeed, invariance criteria should not correspond to given axioms for they would depend on one of the systems that they should simply make possible. Hence, Natorp’s purification of intuition is partial and does not give up the intuitive force of Kant’s mathematical construction, that is, the fact that our intellect must draw––at least mentally––geometrical concepts. One has a sort of concreteness in thought which is different from either merely logical possibility or empirical reality. For this reason, lawfulness is not sufficient to unambiguously provide existence in space, as Natorp’s judgement on Klein’s “ideal abstraction” shows (Natorp 1910: 317). Another major issue with Natorp’s impressive enterprise concerns his references. It seems that, on the one hand, (i) he instrumentally seeks to find works that will corroborate his ideas; on the other hand, (ii) that he deals with adverse sources to simply reinforce his view. The mention of Morton Churchill Mott-Smith (1907) 7 Gödel

(1949: 557–558, foot. n. 3) already drew attention to the fact that this was but Kant’s standpoint (see KrV, B54). 8 The constitutive power of the a priori is a key of Reichenbach’s early neo-Kantianism. See: Reichenbach (1920: 74).

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belongs to the first kind of citations, while the reading of Josef Wellstein to the second. I will succinctly tackle Wellstein’s work, which I find more interesting for our purpose. Wellstein edited his Encyklopädie der Elementar-Mathematik with his master Heinrich Weber. In the second part of the second book, Wellstein tackles “natural geometry” in the light of non-Euclidean geometries and he devotes section §14 to the discussion of Kant’s intuition. In the first place, it is to him clear that natural, to wit, Euclidean geometry is but a possibility among many other options. It follows that the vividity of the Euclidean figures is not a privileged object of knowledge. In addition, Wellstein admits that in the case of “construction” too, one must hold to a strictly deductive proof. A given system of axioms must always fulfil a series of assumptions in respect of which the axioms represent the content (Wellstein 1905: 112). Wellstein thus raises the question which is focal to Natorp: once one considers the relationship between axioms and a possible manifold, is one supposed to find something univocal? Wellstein answers negatively, at least if one starts from Hilbert’s axiomatisation: here one has a whole system of connected definitions and concepts which makes it impossible for us to conceive of any univocal coordination (§§16–17). Even in the case of collineations, the exact one-to-one transformations between points, straight lines and surfaces presuppose a whole series of geometries, so one does not know which is the real Euclidean geometry. They are all likely Euclidean geometries (Wellstein 1905: 118–119; see also: 124). In sum, Wellstein focuses on a new definition of the concept of axiom and introduces some empiric elements in a rigidly idealistic structure. As far as the first point is concerned, it is easy to understand why he also relies on a wider concept of coordination: “Axioms do not effectuate a synthesis, a construction of geometrical figures, but a selection of proper manifolds and connections from the entirety of all which are thinkable” (Wellstein 1905: 120). It is thus clear that one can gain a restriction of logical possibilities only by considering physical geometry. In this respect, Wellstein also upholds that geometric figures are not given in intuition, but through schematism (loc. cit.: 125). The fact that one can represent motions via different concepts and objects means that one cannot leverage merely isometric transformations. That is, one cannot consider the principles of Euclidean geometry as the only ones which would be strictly a priori and those from which we construct geometric structures. One likely consequence is that one should relativise Kant’s a priori forms; at the same time, one can correct the concept of experience and eliminate its state of “given”. In practice, a priori concepts are not fixed ideas; on the other hand, one does not grasp objects that are not logically constructed. It follows that one conceives a priori concepts as conditions of possible experiences, and experience only as something which is possible on the basis of constitutive concepts. Wellstein notices: “Natural science does not observe pure phenomena; rather, it supposes them, to bring observation to laws [ . . . ]. Accordingly, pure phenomena and exact laws remain ideas. To explain nature means to build (aufbauen) it starting from pure phenomena by means of exact laws” (loc. cit.: 142). Therefore, the actual world is a limiting-mathematical structure towards which one progresses by focusing on

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preliminary concepts (Friedman 2001: 68). In general, that would also mean that our mathematical interpretation of natural phenomena is heuristic, that is: one cannot achieve a perfect copy (Abbildung) of facts through mathematics. At this point, Wellstein, who recalls Cohen on the point, takes the floor as a true neo-Kantian: In this way, we can only come to the idea that we must put into effect orders which spring from given grounding orders. In this respect, geometry is a priori, that is: necessary to experience. However, its primary principles cannot be, from the beginning, ensured knowledge, but uniquely hypotheses––in Plato’s meaning––suggested via experience. Therefore, they are approaches that we tentatively make to gain a commencement in general and thereby lay the foundations of relative knowledge. The more hypotheses show to be true, the higher their epistemological value grows. It can happen, however, as physics often teaches, that hypotheses cannot be actualised. [ . . . ] Only when the grounding hypotheses are known as self-consistent and sufficient to the determination of what is real, do they amount to knowledge in the proper sense. [ . . . ] Alongside such a geometry that may work as the most appropriate foundation of mechanics and thereby should be called natural ᾿ geometry κατ᾿ εξoχ ην, ´ there are still other, technical and feasible geometries. If we conceive of the goal of our geometries to be their injection in the union of the whole of our natural knowledge, then their axioms are still today hypotheses. (Wellstein 1905: 145)

Accordingly, it seems that one can avoid any misunderstanding by replacing the concept of a priori with that of conditioning law, a “hypothesis”, which however enables a freer concept of coordination. Clearly, one is addressing a revolution of neo-Kantianism from within, which theoretically paves the way for Cassirer’s readjustements. For his part, Natorp devotes a whole sub-section (sixth chapter, §8: 318–325) to Wellstein. He praises many of Wellstein’s arguments, especially those concerning the criticism of the given and his platonic descriptions that facts should near ideas. He also agrees with him that, theoretically, Euclidean geometry is not the only form of coordination between axioms and experience, and that the integration with physics is required if one is to perform such a coordination. However, he cannot accept Wellstein’s ambiguity when he relies on both Kant’s idealistic claims and empiricism.9 When one asserts that Euclidean geometry is a priori one neither intends it is a “coerciveness” (Zwangsläufigkeit) of our spirit nor that Euclidean space is somewhat preformed in our mind. Rather, one simply assumes that to come to scientific knowledge one needs to coordinate appearances in a certain way. It follows that if one is interested in moving closer to experience, one must shift from the “how” (wie) of a free coordination to the “that” (dass) in which it eventuates (Natorp 1910: 304). It further follows that this passage, as already explained, goes through a Euclidean construction. It is finally important to notice that Natorp’s parlance diverges and will diverge from Cassirer’s. While, as we will see, Cassirer relies on the unambiguousness of coordination, Natorp invokes the “unambiguousness” (Eindeutigkeit) of space and time and the “unambiguous determination of motion”. We have also seen that he often prefers the word “Einzigkeit” to underline the demand of univocality. 9 Interestingly,

empiricism”.

Natorp once spelled out (1900: 379) that in Kant there was a “remainder of

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Table 5.3 The diagram of Natorp’s main theses and their connection. Dashed arrows display problematical links

Table 5.4 The application of Hentschel’s expanded criteria shows that Natorp’s reading is fairly accurate (8/13). As regards point n. 3, we should limit ourselves to alignment with theory (up to 1910). However, the lack of criteria n. 1, 4, 8, 10 makes Natorp’s attempt mostly conservative. That entails the lack of point n. 13

Before seeing what these differences imply, I provide below a map of Natorp’s interpretative theses and hypotheses, together with the Hentschel diagram that one can apply to our findings so far (Tables 5.3 and 5.4).

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I conclude this section by giving a look at Natorp’s further references to relativity. Essentially, in Allgemeine Psychologie one deals with the same stance of Grundlagen (Natorp 1912: 164–165). The last reference to Einstein is to be found in Philosophische Systematik, where Natorp argues in favour of a topological conception of space, which would precede the purely metric one (Natorp 1958: 265–266; Ferrari 1996: 132, foot. n. 86). Arguing for the aprioricity of topology is another way to preserve Kantianism from relativity.

5.3 Cassirer If one can say that scholars have devoted sufficient attention to Cassirer’s interpretation of relativity, it is no less true that scarce emphasis has been placed on how it binds with Natorp’s book. In the second section of this paper, my aim is to delve into this matter. To do so, I will briefly outline Cassirer’s interactions with his mentor and introduce Schlick’s and Einstein’s receptions of Natorp’s reading. Then, I will focus on Cassirer’s correspondence on relativity (mainly with Einstein and Schlick) and his general interpretation of the theory, with special emphasis on the comparison with Natorp. Finally, I will briefly discuss possible neo-Kantian reverberations in Einstein’s epistemology.

5.3.1 Cassirer’s Reaction to Natorp’s Book Cassirer’s reaction to Natorp’s book does not come out of the blue. Certain letters shine light on what happened. But first, one must consider that Cassirer was debating the nature of mathematical knowledge with Russell and Couturat between 1906 and 1907. His early philosophy of mathematics is characterised by the commixture of formalist and logicist claims with the defence of the synthetic meaning of mathematical propositions. This created in his mind the idea of a possible “agreement” (Einklang) with the new generation of mathematicians (Cassirer to Natorp, 3 June 1906: Holzhey 1986: 346–348). I imagine this to be an immunising approach since Cassirer believes that the convincement should stem from the acknowledgement by Russell and others of incorrect interpretation of Kant. However, Cassirer himself discovers issues in this defence. In a letter sent to Natorp on 28 June 1906, Cassirer states that mathematical concepts are but “norms of thought”, viz. “relations”. This entails that they rest in their “unambiguousness” (Eindeutigkeit) and not in their capacity to reproduce “facts”. He also recognises that “intuition” is not important foundationally, but in a regulative sense since it helps us to choose among different axiomatic systems, which are all equally possible. Moreover, he is not sure that this is Natorp’s opinion, and asks him for clarification. Cassirer’s question is revealing: how does one differentiate between a “space-” and a “time-point” without intuition?

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Natorp’s answer is to be found in a diagram on p. 344 of his Grundlagen, where he suggests that one can reduce all existence to point-coincidence in a Euclidean manifold defined by space- and time-coordinates. While in time one hinges, indeed, on the “differentiation” (Sonderung) that occurs within the process of generating values by following a rule, in space one focuses on the “connection” (Verbindung) of points. Therefore, space makes the time-distinctions effective in experience since it enables the mutual comparison of time-like points; time, for its part, endows the absolute homogeneity of space with a sort of principium individuationis that allows us to distinguish qualitatively between each space-like point. Now, Natorp recognises that in principle one can generate countless frames by picking any point of the manifold as the origin, but he underlines that the passage to physics entails a further determination. In short, Natorp maintains that in mathematics one does not truly tackle any space or time, but only numbers, while space and time are objects of mechanics (Natorp 1910: 279). That Cassirer is not satisfied is clear when reading the letter he sent on 30 October 1909, in which he reviewed his master’s book. This letter contains many remarks which I cannot address in full. Nevertheless, it is striking, on the one hand, that Cassirer criticises Natorp when the logical and ordinal meaning of mathematical structures is called into question; on the other, that his main concerns revolve around the sixth chapter of Natorp’s work, which is entitled “Zeit und Raum als mathematische Gebilde”. Cassirer agrees with Natorp’s assessment of Wellstein, but only “relatively” and apart from “some misunderstandings as to the ‘transcendental’ question” (Cassirer to Natorp, 30 October 1909, Holzhey: 385). In accordance with his master, he excludes that mathematicians should endorse empiricism because of meta-geometry; quite the contrary, meta-geometry proves that geometry is a priori. However, there is an incendiary comment pertaining to the judgement of existence: according to Cassirer, there is no justification for making of the space of intuition a 3d manifold. Hence, despite the aprioricity of geometry, there is no a priori geometry.10 In general, Cassirer thus emphasises more strongly the highly deductive character of geometric structures (see Schiemer 2018; Biagioli 2020; Reck 2020, among others) and leans towards a more liberal account of the concept of coordination. In this respect, suffice it to say here that the pivotal role played by Klein’s program in Cassirer’s Substanzbegriff und Funktionsbegriff speaks volumes as to the difference from Natorp. The reality or existence of space is a foreign concept to Cassirer, who conversely conceives of geometry, at least in this phase, as an attempt at defining a hierarchical set of norms and operations conditioning a given arrangement of the manifold (Biagioli 2016: 201). Yet the surprises are not over. Indeed, Cassirer endorsed Wellstein’s idealistic stances but rejected his complaint about Kant’s empiricism (1953: 106 and ff.). Furthermore, perhaps rather paradoxically, in 1910 Cassirer’s opinion was still in line with his master’s about the primacy of Euclidean

10 Natorp wrote back to Cassirer on 4 November 1909, but unfortunately I have not been able to decipher Natorp’s handwriting in this case.

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geometry, for its being the simplest and most abstract one (loc. cit.: 109), although he clearly manifests a penchant for a formalist and, as said, structuralist description of geometrical knowledge (Ihmig 2001: 4 et passim; Biagioli 2016: 200–201). Finally, if one wonders why Cassirer did not tackle relativity theory in Substanzbegriff und Funktionsbegriff, one may find the answer in a letter Cassirer sent to Natorp on 28 November 1909, where he states that his knowledge of the most recent physics is not as in-depth as would be necessary to study relativity. He nonetheless promises to delve into the subject soon. In a letter dated 16 May 1911, Cassirer says that he encouraged Otto Buek to work on the “relativity principle”, which may bear witness to the inception of his work on Einstein. From this point forward, however, he appears to have postponed his own project for many years. He mentions his book on relativity to Natorp only in a letter dated 9 September 1920, when he is already in touch with Einstein.

5.3.2 Schlick’s and Einstein’s Reactions to Natorp’s Book Before addressing Cassirer’s interpretation, it is interesting to ponder over Schlick’s and Einstein’s reviews of Natorp since empiricism was triumphant in the philosophical debate on relativity at that time.11 Schlick’s review was pungent. The leader of empiricism dealt with Die logischen Grundlagen in 1911 and again in the sixth section of his Die philosophische Bedeutung des Relativitätsprinzips (1915). Essentially, Schlick underscored two main issues. In the first place, he suggested that the idealistic nature of absolute space and time also refers to their empiric measurements. In the case of empiric measurements, one needs ideal criteria to provide the comparison between bodies upon which we lay a rod as well as to compare motions or other processes. It follows that Natorp had too narrow an idea of empiricity. In the second place, Schlick vindicated that relativity theory deals with an “eindeutiges Bild des Geschehens” without the need for absolute space and time. That would indeed imply a dogmatic restriction, which also affects the definition of the scientific object as something one asymptotically nears to the infinite (Schlick 1915: 33–39). Einstein briefly addresses Natorp’s conception of space and time later in his review of Elsbach. Essentially, Einstein finds Natorp’s exposition to be “transparent”, though it put the essential aside. He admits that if one cannot test geometry empirically, one is not automatically forced to assume the “ideality” of space and time. Indeed, space and time are concepts which share ideality with all other notions. Thus, one cannot test geometry for its ideality, but only for being a part of a whole

11 Truth be told, nowadays we know that all we can state of Einstein as an epistemologist is that he had always been a holist. Indeed, Howard and Giovanelli (2019) showed that Einstein’s preference for holism dated back to 1910 or 1911, and was prevalent since the turn of the 1920s. An explicit statement is given by Einstein in The World as I See It (1935: 172).

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system of concepts which is selected “for reasons of simplicity and practicality” (Einstein 1924b: 1691). However, I do not agree with Einstein when he says that the ideality of space is not meant by the philosopher as the freedom of choice concerning the coordinate system. Theoretically at least, as we have seen, Natorp distinguished the logical possibility of multiple forms of coordination from the real possibility of Euclidean geometry as to the judgement of existence: it is only in this second sense that one is forced to go through Euclidean geometry. As paradoxical as it may sound, Natorp likely wanted to underline that despite the fact that one can save appearances in multiple ways, when their geometric representation is necessary, one is compelled to start from the construction of geometric structures in an almost unitary intuition. As said, this was acceptable in 1910. Moreover, one should consider that Einstein may have been familiar with some of the Marburg philosophy. Earlier, I mentioned Otto Buek, who completed his doctorate under Cohen and became friends with Einstein. It is thus possible that the latter heard of the discussion about the concept of coordination and its nature also with respect to its neo-Kantian version (Howard 1992). However, it is no less striking that in 1924 Einstein still desired to take a stand against its more conservative form.

5.3.3 The Reason of “Logos Itself”: Cassirer’s Correspondence Although I am aware that the correspondence is not sufficient to shed light on the philosophical issues of relativity, it is nonetheless useful to understand certain comparisons by extracting from letters those judgments which are less influenced by a certain degree of academic prudence. As far as relativity theory is concerned, one may list the following items: 1. 2. 3. 4. 5. 6. 7.

Cassirer to Einstein, 10 May 1920; Einstein to Cassirer, 5 June 1920; Cassirer to Einstein, 16 June 1920; Cassirer to Natorp, 15 October 1920; Cassirer to Schlick, 23 October 1920; Cassirer to Reichenbach, 27 April 1922; Einstein to Cassirer, allegedly 1924.

Some of these letters are well-known among Cassirer’s scholars (Ferrari 1996); nevertheless, it may be useful to sum them up shortly and to delve into a couple of important passages. In (1) Cassirer confessed that he did not want to publish his personal studies on relativity, perhaps because he found the enterprise overwhelming. However, he felt the importance of fostering the dialogue between physicists and philosophers and thus asked Einstein to review his book. In (2) Einstein wrote back to Cassirer with his comments. While generally praising Cassirer’s endeavour, he nonetheless raised two heavy criticisms. First, he recognised that to approach

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experience one needs “conceptual functions”, but he points out that the choice is never conditioned by “the nature of our intellect” (CBW: 46). Second, he underscored the fact that to accomplish the coordination between axioms and experience one must clarify the “way” in which it happens. Indeed, at the turn of 1920s Einstein is convinced that coordination is enabled by an empiristic intermediation (Howard and Giovanelli 2019). With this principle, Einstein cast a shadow on Cassirer’s analysis of ds and stated that “GR depends on the interpretation of ds as measure outcome, which has to be gained through rods and clocks in an utterly determined way” (CBW: 46). In (3) Cassirer acknowledged that Einstein had compelled him to revisit his theses and to consider more seriously “the purely empirical origin of relativity theory” (CBW: 47). Nevertheless, this is in contrast with (4), whereby Cassirer fell in line with his master’s idealistic interpretation. He also begged him to develop the conclusive sections of Die logischen Grundlagen to transpose the discussion to the realm of “pure philosophy”. But perhaps more importantly, Cassirer confessed all of his happiness that Einstein had given him positive feedback about his comprehension of the theory. Therefore, even though he was aware that as regards “the epistemological consequences” it was hard to agree with him in full, in virtue of his essentially empiristic vision of physical geometry, he seemed optimistic that, thanks to their interaction, Einstein was now closer to the neo-Kantian standpoint. The fifth letter (5) displays how these opposite stances are peculiarly meshed. Cassirer faces here the pioneer of empiristic philosophy. Howard and Giovanelli (2019) have recently clarified that Einstein struggled with empiricist philosophers at the very beginning of the 1920s, when they tried to present Einstein’s concept of “convention” as the very one they defended. It is thus noteworthy that in this place Cassirer both liberalised Natorp’s immunisation and explained why the concept of the “unambiguousness of coordination” (Eindeutigkeit der Zuordnung) does not fall under Schlick’s conventionalism (Schlick 1918: 312 and ff.; Id. 1921: 142– 143). First, Cassirer pointed out that his concept of a priori eventuates in neither intuition nor concepts since it is “a function”, that is, a law that establishes patterns or forms whose content varies throughout history. Second, he thus considered as a priori only that “unambiguousness of coordination”, which eventually amounts to “an expression of ‘reason’, of logos as such” (CBW: 51). Schlick opposed this standpoint in his review Kritizistische oder empiristische Deutung der neuen Physik?: he thought that Cassirer had simply drawn attention to a quite obvious definition of physical law (Schlick 1921: 131 and ff.). However, Cassirer believed that he concurred with Einstein for having upheld that “principles” or “premises” are the result of the history of science. In fact, Einstein believed that every choice concerning a set of concepts to be established as a priori is arbitrary (Einstein 1924b: 1689–1690) but not that it is impossible to deal with a priori structures. Of course, an arbitrary distinction between a priori and a posteriori would be inane from Einstein’s standpoint (Howard 1994: 91), thus one may ask whether Cassirer’s position is sound. Be that as it may, the content of the a priori is not fixed in either cases (Friedman 2001: 79–92, 2008; Ibongu 2011: 29). Therefore, Cassirer reveals in (6) that his account deviates only terminologically from that of Einstein as a spokesperson of

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the neo-empiristic tradition: “As far as the difference from Petzold is concerned, I agree with you and also believe that Einstein’s observation rests on the fact that he does not diverge from me objectively, but terminologically”. In (7) Einstein confessed to Cassirer his doubts about considering causality as an a priori law of nature. He said, with sarcasm, that a Kantian philosopher would argue that our perceptions answer a priori representations which order exactly those perceptions. Accordingly, such representations would not stem from experience and not fit in with a principle-theory, the only allowing inductive generalisations. Considering Einstein’s tone, he finds the Kantian presupposition circular, if not foolish (CBW: 62–63). Perhaps he was simply recalling Schlick’s criticism that Cassirer’s a priori is but the definition of law, that is, an analytic principle that establishes the possibility of tying up relata with one another. This brief inquiry into Cassirer’s correspondence has shown that he is disposed to revise Kantianism profoundly, but that he could not quit conceiving of an a priori level which, so to speak, prepares the empiricity of physical laws. Let us see how his interpretation of relativity swings between these two trends.

5.3.4 Cassirer’s Interpretation of Relativity Apart from the correspondence, one can count these main contributions by Cassirer on relativity theory: 1. A paper that appeared in “Die Neue Rundschau”: Philosophische Probleme der Relativitätstheorie (1920); 2. The text of the winter semester class 1920–1921: Die philosophischen Probleme der Relativitätstheorie; 3. The book: Zur Einsteinschen Relativitätstheorie. Erkenntnistheoretische Betrachtungen (1921); 4. The fifth chapter of the third volume of Philosophy of Symbolic Forms (1927/1929); 5. The rectorship’s speech in Hamburg (1929); (1), (2) and (3) are chronologically interwoven and reveal no essential contradiction, though in the class Cassirer gets, as it were, more philosophical and makes an interesting reference to Hegel––as well as in (1).12 (4) and (5) are essentially retrospective, but involve an important assessment of Eddington’s interpretation. Needless to say, Cassirer’s references to the literature on relativity were impressive. The reference list of (3) contains more or less all the essentials up to 1920, including Poincaré, Duhem, Lorentz, Ehrenfest, Erwin Freundlich, and von Laue; that of (2) also includes Planck, Einstein’s Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie, and Born’s book on relativity theory. Philosophi-

12 Among other topics, Cassirer refers to the fact that from the empirical standpoint the “thing” can be conceived of as a “category” (1920: 226).

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cally, Cassirer recommends: Frischeisen-Köhler, Hönigswald, Natorp, Sellien, and Petzoldt. In (3) Cassirer picks out the classics on SR and GR from 1905 and 1916, together with a brief celebrative paper on Mach (1916b). Surprisingly, Cassirer does not mention Schlick’s first paper on relativity. It is hard to explain why: perhaps Cassirer thought that Raum und Zeit in der gegenwärtigen Physik (1917) and the first edition of the Allgemeine Erkenntnislehre (1918) were representative of Schlick’s standpoint. Cassirer does not even refer to Hilbert. Of course, to delve integrally into such a network of topics and authors is far beyond the scope of this paper. Furthermore, (4) and (5) would compel us to abandon the history of epistemology and embrace metaphysical issues that reverberate in the philosophy of culture.13 I will thus confine myself, on the one hand, to pursuing the objective of a comparison with Natorp and sketch Cassirer’s interpretation as it stems from the one provided by his master; on the other, to reflecting on possible inconsistencies and the philosophical development of Cassirer’s standpoint. I will discuss the latter topic in the next section. Natorp is generally spoken of in three ways. Firstly, (i) there are unmodified argumentations. Secondly, (ii) one encounters references which only apparently align with Natorp. Lastly, (iii) Cassirer gives a nod to other standpoints, to such an extent that some scholars hold that “there is not enough left in Cassirer’s Kantianism for it to deserve to be called Kantian” (Heis 2011: 791). A mention of the first kind regards the so-called Grenzbegriffe. The speed of light, for instance, is a condition of possibility exactly in the sense explained by Natorp. However, Cassirer notices that such a condition is different from the more general constraints introduced in GR. The speed of light is still a material a priori and does not fulfil the idealisation required by the progress of physical theory. Indeed, it is known that Cassirer captured the transition from SR to GR by stating that one shifts from “material” to entirely “formal” principles (Cassirer 1953: 377 and ff.). Therefore, Cassirer draws attention to a peculiar distinction between purely formal and content concepts. He claims that space and time as pure forms of order are invariant with regard to all changes in physical theory; nevertheless, contents are not merely separated in the sense that one firmly distinguishes the transcendental from the empirical part. In fact, modifications in the relativisation of coordinates impacting the theory are not merely empirical. All of this naturally leads us to the second kind of argument. Indeed, Cassirer is aware that to make the new vision compatible with Kantianism, the key rests on the point-coincidence in a 4d manifold. The fact that one requires a coordination in general between axioms and experience is thus a priori in the sense that one cannot expect less than the notion of interval in geometry; on the other hand, it is unnecessary, if not completely wrong, to invoke the primacy of  Euclidean geometry in this respect. This becomes clear if one focuses on (4): .ds 2 = 4μ,ν=1 gμν dx μ dx ν

13 However, it is worthwhile mentioning that Ryckman has already elucidated the momentous import of relativity for the elaboration of the philosophy of symbolic forms in one of his seminal papers (Ryckman 1999: 614).

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(Einstein 1916a: 819). As suggested by Ibongu (2011: 22 et passim), Cassirer correctly infers that it accounts for the background independence since one has a spatio-temporal continuum that changes locally (Torretti 1996: 101; Rovelli 2021: 34 and ff.). Two things thus follow: there are no privileged frames of reference in the description of relative motions; and the world is given, according to Weyl, in a 4d metric (Weyl 1919: 244; Cassirer 1953: 398). At this point, considering Natorp’s attempt at collocating the neo-Kantians within the sphere of influence of Poincaré’s conventionalism, the question becomes how to negotiate between “imposition” and arbitrariness in virtue of the weight of experience in the choice of axioms (Poincaré 1908: 5; 64–67). To Natorp, Poincaré merely seems to be useful for defending the difference between geometry and experience, and thus served the purpose of maintaining a transcendental status for mathematical construction. I suggest that Cassirer solved the puzzle by replacing imposition with spontaneity as the very source of coordination. This reinforces the strategy involved in (ii) and lets us jump to (iii). Let us begin by adding some considerations about the terms denoting the act of coordination. It is noteworthy that Natorp mostly employs “eindeutig” as an adjective for “Zuordnung”, while Cassirer undoubtedly prefers “Eindeutigkeit” as a noun. In fact, as we have seen, Natorp singles out Euclidean over non-Euclidean geometries in terms of construction; Cassirer, for his part, while apparently espousing his master’s orthodoxy in 1910, gives up any rigidly Kantian concept of intuition in his book on Einstein (Cassirer 1953: 418). It is thus true that he emended his work after Einstein’s review. In Cassirer’s epistemology, the concepts of function and invariance have overwhelmed any other canonical reference to Kant, to the point that even the quotation of Kant’s dissertation is leveraged to ground the concept of ds. Therefore, as said, the relational and structuralist conception of mathematics sets up a freer coordinative ideal which, by rejecting any foundational character for Euclidean geometry, will allow Cassirer to simply demand that a set of hypotheses agrees with experience and that such an accord is univocal and as simple as possible each time. If this does not resemble an empiristic vision on coordination, I do not see what else might.14 However, we have also seen that in his letter to Schlick, Cassirer defended the existence of a priori in another sense. He is convinced that despite the fact that the coordination between axioms and experience is empirical, it is nevertheless the exigency of a coordination as such to be retained as a priori. This sounds more or less like Leibniz’s objection to Locke: one can go forward as much as one wishes in arguing that the mind is a tabula rasa and that cognitions hinge on experiences, but the image of the tabula is prior to experience (Gerhardt 1882: 99 and ff.). By the same token, if one aims to coordinate axioms with experience, such a coordination should be possible in general before it can become actual. Hence, Cassirer holds to the distinction between logical and real possibility (Kant 1797), though unlike

14 In a private note, Cassirer upheld that he stands “closer to no other philosophical ‘school’ than to the thinkers of the Vienna Circle” (Ibongu 2011: 57).

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Natorp he does not go through any Euclidean construction. To understand how this is accomplished in concreto, let us take a look at Cassirer’s reading of general covariance. This topic has been discussed masterfully by Ryckman (1999, 2005), but it is worthwhile to address it anew for our purpose. To begin with, we may distinguish passive from active general covariance. The former consists of a coordinate transformation from S to S and changes the labels of the points of the two systems; the latter focuses on the points and proves that different metrics are fit for the same points. In the first case, we see the same event from different perspectives; in the second case, metrics actively generate more solutions for a given transformation (Ryckman 2005: 20). For his part, Cassirer sides with the freedom of choice in changing coordinates (Cassirer 1953: 441 and ff.). I imagine that he mostly means freedom of coordination (Biagioli 2016: 205). That he has taken general covariance seriously is shown by his paper on relativity, where he realises that, formally, the form of space does not differ from that of time (1920: 228), as well as that the concept of point-coincidence plays a prominent role. Indeed, the latter was pivotal to allowing Einstein to overcome the fallacies of the hole-argument, which entails active general covariance (Pais 1982; Norton 1993; Ryckman 1999; Giovanelli 2021). Several of Cassirer’s quotations point in this direction.15 First, in the class he states: “Following the grounding postulate of relativity theory, all arbitrary coordinate systems are equivalent and have equal right for the description of natural phenomena and the setup of their laws” (1920/1921: 41). Therefore, in making of general covariance a heuristic principle (Cassirer 1953: 377; Ryckman 1999: 604; Ihmig 2001: 158–174), Cassirer believes he has reached his goal, that is, to have idealised all matter: “We keep a curve in the world, a worldline, then, as an image, as it were, of the constant story (Lebenslauf ) of the substantial point––the entire world appearing solved into such worldlines and all physical laws expressing but correlations between these worldlines” (Cassirer 1920/1921: 114). Such an idealisation radicalises the original project. It is possible that Cassirer behaved prudently until he discovered a physical theory which would reinforce his standpoint, so he remained cautious and respectful of his master’s ideas, until he could advance more grounded statements. Therefore, if from the beginning Cassirer’s aim was that to remain, so to speak, alone with laws, it is only general covariance that could work as the final proof that idealisation does not merely seal mathematics, but also physics. To a certain extent, Cassirer’s epistemology of relativity is thereby a philosophical translation of Einstein’s purpose to efface any remainder of physical objectivity16 in the representation of spacetime (Einstein

15 It is also noteworthy that in these passages Cassirer openly endorses Duhem’s conventionalist account of ancient astronomy (loc. cit.: 40–41). About Duhem’s influence on Cassirer, see: Itzkoff 1971: 57–64, 74; Ihmig 2001: 102–126; Schmitz-Rigal 2002: 220–225; Ferrari 2015: 17–18; Richardson 2015. 16 In The World As I See It, Einstein notices, however, that space and time are “divested not of their reality but of their causal absoluteness” (1935: 155).

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to Schlick, December 14, 1915: CP 8A, doc. 165; Einstein 1916a: 776). Therefore, passages like the following are intended as a direct prosecution of Einstein’s project: And now we can ask, epistemologically, only one question: whether there can be established an exact relation and coordination between the symbols of non-Euclidean geometry and the empirical manifold of spatio-temporal “events”. If physics answers this question affirmatively, then epistemology has no ground for answering it negatively. For the “a priori” of space that it affirms as the condition of every physical theory involves, as has been seen, no assertion concerning any definite particular structure of space in itself, but is concerned only with that function of “spatiality” in general, that is expressed even in the general concept of the linear element ds as such, quite without regard to its character in detail. (Cassirer 1953: 432–433)

It is clear that Cassirer’s interpretation is deeply grounded in his earlier work: in fact, the starting point is the transcendental configuration of geometry, which is in no way comparable with the classical concept of construction. To make a geometry possible suffice it to get relations of coincidence between points-events, on the basis of the analytic conception of continuity. It is only in this way that the possibility of coexistence becomes the “non-deductible fundamental concept” of all geometries (Cassirer 1957b: 43 and ff.), and that geometry as such is a priori. Therefore, applying mathematical concepts to empiric descriptions does not imply merely injecting fixed a priori forms into experience. Rather, empiric forms should be generated starting from theoretical structures. Besides, already in 1910, Cassirer was convinced that physics submits to us manifolds which require “a plurality of means of determination” (Cassirer 1953: 110). In some respects, Schlick is thus right (1921: 140–141): Cassirer is beyond the field of criticism,17 and is ready to account for Einstein’s demand that the structure of spacetime cannot be addressed (only) topologically (Cassirer 1953: 425; Einstein 1920: 65–67; Ryckman 1999: 593 and ff.; Ihmig 2001: 166). As a consequence, Cassirer does not pretend that empiricity must be absolutely determined. Quite the opposite, he leaves room for physics to arrange physical tests of geometry, while affirming that it is not in a position to decide which geometry is “real”. In this respect, I do not find any discrepancy from Einstein. Incidentally, Cassirer also highlights that simplicity, homogeneity and continuity are necessary criteria to determine theory, which is exactly the advantage of relativity as a principle-theory: a preference for “logical perfection and security of the foundations” (Einstein 1935: 168). However, that also leads Cassirer to label covariance as an analytic and “almost arbitrary” principle (Cassirer 1953: 366). Thus unsurprisingly, in the class, Cassirer refers to the principle of relativity as the “law of lawfulness”, an expression he borrows from Natorp (Cassirer 1920/1921: 88). Such a peculiar combination of claims may lead one to ask oneself why to compare Cassirer’s transcendentalism with Einstein’s principle-theory, which 17 In the first chapter of the book, Cassirer mentions the questions that were addressed by neoempiricists, especially as regards the commitment of Kant’s philosophy to Newtonian mechanics, and admits that we have to move beyond Kant (Cassirer 1953: 355–356; Ibongu 2011: 29 and ff.).

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essentially depends on inductive generalisations of natural processes (Einstein 1935: 166–173; Hentschel 1990: 15 and ff.: Howard and Giovanelli 2019). The answer is that in principle-theories empirical laws must obey more general principles specifying the rules under which those laws are to be constructed. Einstein pinpointed, indeed, that relativity theory originated from “experimental necessities” (Einstein 1922: 97). Hence, if Cassirer’s “physics of principles” does not fit Einstein’s verve in epistemology, it is nonetheless structurally equivalent to it (Giovanelli 2016). All of these seemingly antithetical statements can thus coexist if one bears in mind that the act of coordinating is transcendental, while a given coordination is not. In a nutshell, transcendental principles are not metaphysically prior to experience (Itzkoff 1971: 78; Ryckman 1999: 601), but work to let it manifest in an ordered fashion. Accordingly, Cassirer’s strategy eventuates in a liberal development of Natorp’s position, which also stands for a defence of what neo-Kantianism can reasonably be in virtue of relativity. Considering that relativity has in fact modified Kantian philosophy, Cassirer acknowledges that Einstein’s theory reaches at least a first meta-empirical level since it reflects on the conditions of possible space- and timemeasurements. He notices: Thus, physics knows its fundamental concepts never as logical “things in themselves”, but only in their reciprocal combination; it must, however, be open to epistemology to analyse this product into its particular factors. It thus cannot admit the proposition that the meaning of a concept is identical with its concrete application, but it will conversely insist that this meaning must be already established before any application can be made. Accordingly, the thought of space and time in their meaning as connecting forms of order is not first created by measurement but is only more closely defined and given a definite content. We must have grasped the concept of the “event” as something spatio-temporal, we must have understood the meaning expressed in it, before we can ask as to the coincidence of events and seek to establish it by special methods of measurement. (Cassirer 1953: 420)

Truth be told, this passage was prepared earlier in the book: The property of not being thing-concepts, but pure concepts of measurement, space and time share with all other genuine physical concepts; if, in contrast to these, space and time are also to have a special logical position, it must be shown that they are removed in the same direction as these, a step further from the ordinary thing-concepts, and that they thus represent, to a certain extent, concepts and forms of measurement of an order higher than the first order. (Cassirer 1953: 357–358)

Although there is a technical problem with the attempt at combining the arbitrariness of coordinate transformations with the realisation of a group (Torretti 1996: 154), Cassirer’s approach is clear. Just as one can incorporate geometries in a vaster framework through which one focuses on their different status in light of the generality of their axioms and replaceability of the operations determining them (Biagioli 2020: 37), it is possible to conceive of two interrelated levels of which only one is strictly empirical. But, considering that one only addresses different relation degrees (see also Cassirer 1927: in part. 46–66), “lawfulness” and “inclusivity”, and not intuition or “existence” are the keys of Cassirer’s epistemology (Itzkoff 1971: 74 and ff.).

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Perhaps aware of the potential lack of empiricism that he is conversely said to have embodied in his work, and conscious that it was necessary to align with Einstein, Cassirer attempts to fix the glitch with a reference to Kant’s excerpt on space as commercium (Cassirer 1953: 411 and ff.). In particular, one begins with an assertion upon the relations of things and, afterwards, outlines general laws that are shown to be preliminarily grounded in our concepts of space and time.18 I suspect this to be a petitio principii since Cassirer concludes that the “unambiguousness of coordination” stands for the aprioricity of space and time (loc. cit.: 415), but it bears witness to Cassirer’s will of agreeing with Einstein as much as possible while striving to remain a neo-Kantian. That Cassirer could not accomplish either of the two tasks does not undermine his intent. Before entering the final sub-section, I wish to draw attention to a circumstance emphasised by Howard (1997). Einstein conceived of the “Eindeutigkeit der Zuordnung” as the successful coordination of a point of the continuum with a set of numbers which define its dimensions, as well as the infinitesimal intervals with other points (Einstein 1920: 61). As a reader of Schopenhauer, he may have been influenced, perhaps unwittingly, by the vision according to which space and time are principles of individuation (Schopenhauer 1912: 141 et passim). Howard notices that the only philosopher to directly refer to the topic is Cassirer in Determinismus und Indeterminismus (Cassirer 1937: 224–225). For his part, in the Philosophische Systematik (1958), Natorp refers to space and time as “categories of individuation” (Ferrari 1996: 128). Interestingly, for Schopenhauer the concept of intuition in geometry leaves nothing undetermined (Schopenhauer 1891: 151–157), similarly to Natorp’s judgement of existence. It is nonetheless hard to decide whether this mention of Schopenhauer is more than a clue of either a direct influence or a plain agreement between Einstein and the neo-Kantians.

5.3.5 In Search of Inconsistencies: A Final Assessment of Cassirer’s Interpretation In this subsection, I will address the flaws of Cassirer’s reading, but I will also broach its justification. Finally, I will underline why some of Cassirer’s claims may have been suggestive for Einstein. There are two fundamental issues with Cassirer’s standpoint. On the one hand, Pecere (2007) has shown that the idealisation of matter does not fully correspond with Einstein’s vision since the latter may endorse a monistic ontology of the field,19

18 It is also striking that Kant himself referred to the propagation of light as the condition of possibility of action-at-a distance––or of infinite velocity for gravity (KrV, A213/B260, Engl. Tr. 318). 19 Significantly, Ryckman (2017: 239–240) has drawn attention to Einstein’s recovery of the concept of ether, although he conceived of it in a more dynamical fashion.

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furthermore entailing realistic claims about the actual curvature of space and the altered frequency of clocks (Pais 1982). To a certain extent, Cassirer recognises this point (Cassirer 1953: 396–397), but his anti-substantivalism prevents him from permanently embedding in his philosophy empiristic or at least ontological statements.20 However, to interpret general covariance as a heuristic principle mirrors Einstein’s odyssey concerning the proof that general covariance may not only entail the principle of relativity, but remove the last remnant of objectivity as to the representation of space and time (Ihmig 2001: 145–158). It is questionable to interpret this as a formalist statement for relativistic equations involve information about the distribution of mass and energy, but it is nonetheless possible to do so. In way of another example, in a 1930 paper for The Yale Universitary Library Gazette, Einstein affirmed that a non-relativistic theory involves statements about things, while a relativistic theory is purely formal (Pais 1982, chapt. 15). How could be Cassirer entirely wrong on the formalist spirit of relativity? On the other hand, the distinction between general covariance and the principle of relativity is a well-known issue in the epistemology of relativity (see at least Norton 1993), which also impacts Cassirer’s work. In this respect, Ibongu (2011: 50, foot. n. 212) has explained that a general relativistic metric involves more than coexistence and succession, that is, more than a neo-Kantian reworking of general covariance as a relativity principle. However, one must consider three things. First, Cassirer was not aware of the hole-argument (Ihmig 2001: 158). Second (Ryckman 2017: 241, foot. n. 8), there were already seemingly idealistic presentations of general covariance, like that of Hilbert (although Cassirer does not refer to him openly). Incidentally, Hilbert approved the concept of a priori as the class of the “indispensable preconditions” of experience (Giovanelli 2016: 146), and Cassirer wrote that the purely ordinal mathematical determinations “predicate” (behaupten) and “ground” (begründen) the world of “sensitive things” (1920: 219–220). Third, Cassirer’s posterior works show a more prominent treatment of the concept of field.21 He explains, indeed, that in GR all dynamical relations trace back to metric (1925: 449), just as matter is an “Ausgeburt des Feldes” (1927: 35, 1957a: 466 and ff.). In the Philosophy of Symbolic Forms, moreover, physical knowledge is but “symbolische Konstruktion”, realising the unified description of space, time, and matter (1957a: 468). In Determinismus und Indeterminismus (Cassirer 1937: 162 and ff.), the notion of function also relates to Weyl’s conception of field since causality, in GR, concerns “observable facts” (loc. cit.: 156–157). Quite the opposite, in the discussion appended to Cassirer (1931: 428 and ff.), stating that

20 This will be done later as regards the interpretation of quantum mechanics. See in particular the description of field as “omnipresence”, which is not by chance presented as overcoming both Natorp’s spatiotemporal judgement of existence and Schlick’s temporal determination of physical reality (Cassirer 1937: 235 and ff.). 21 I will level in the following the difference between Einstein’s general covariance and Weyl’s gauge invariance to simply highlight that Cassirer was aware that it is impossible to utterly reject any ontological claim about the existence of fields. Regarding the comparison between Einstein and Weyl, see: Ryckman 2005, chap. 4.

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space is a “Maßfunktion”, he rather vindicates aprioricity for space as a general ordering concept.22 Therefore, this confirms that Cassirer is an informed and flexible reader who does not give up, however, his purpose of salvaging aprioricity. Perhaps by bearing in mind that from a physical standpoint conventionalism would be more suitable to account for a liberal coordination of axioms with experience, Cassirer hence defends aprioricity as to the historical development of theories. In practice, the conception of theories as limiting cases of wider to-be-shaped theories balances the lack of apodicticity one experiences through the relativisation of the a Priori. These are words from Cassirer’s class: “If we depict the task of physics in this sense, we see that, despite the constant change of ‘images’ and models, the authentic physical hypotheses are not under threat– –rather, the foregoing hypotheses are ‘sublated’ into the subsequent ones (in Hegel’s double meaning)” (Cassirer 1920/1921: 65). Cassirer’s “quasi-Hegelian” developmentalism in the epistemology of physics was introduced by Ryckman (1999: 599), who also recognised that it is a valuable substitute for a canonical transcendental philosophy. Later Heis (2015: 126) even coined the above-mentioned term “developmentalism”––but surprisingly without mentioning Hegel. To sum up the developmentalist stance, history provides scientific progress with increasingly inclusive and therefore transcendental theories generating particularised contents. As to Einstein’s philosophy of science, I cannot go into detail since, as is known, Einstein was reluctant to accept standardisation (Einstein to Study, September 25, 1918, CP 8B, doc. 624). His epistemology is a multifaceted one that brings together ideas from very different trends (Howard and Giovanelli 2019). I thus limit myself to drawing the attention of the reader to a couple of passages. In the renowned discussion at the Société française de Philosophie (1922), Einstein had the chance to reject Kantianism. Brunschvicg offered him this opportunity with respect to the distinction between mathematics and physics but Einstein did not seize the moment. In addressing the question of the a priori, though vindicating that relativity did not fully agree with Kant’s philosophy, Einstein was not rough. He said that apriorism and conventionalism side with the presumption that science needs “arbitrary concepts”, as one would expect from his letters to Cassirer, but he could not say whether they are given a priori or are conventions: “Quant à savoir si ces concepts sont donnés a priori, ou sont des conventions arbitraires, je ne puis rien dire” (Einstein 1922: CP 13, doc. 131). Although this could be interpreted as a statement of philosophical disinterest, it is noteworthy that some years later, in his reply to the authors who contributed to Schilpp’s volume (1949: 674 and ff.), he more or less leveraged the same argument. Einstein will uphold, like Kant, that he accepts the usage of “categories”; unlike him, nonetheless, he is not convinced that they are immutable. His categories are “free conventions”, useful for interpreting empirical data, which are, in turn, not given simply (loc. cit.: 680). Even though he did not speak on behalf of the neo-

22 By

the same token, it is noteworthy that Hentschel (1987: 466–467) sees idealisation also in Einstein’s theoretical holism, and Rovelli explains that relativistic space is relational and discloses “the order of who is around whom” (Rovelli 2019: 17; see also Ryckman 2017: 224).

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Kantian tradition, he relies on the fact that: “The real is not given to us, but put to us (aufgegeben) (by way of a riddle)” (see also Natorp 1910: 16–22). Furthermore, he mentions the “coordination” as the real substitute of Schlick’s verification.23 For the sake of continuity between these two moments, one can also consider that in France, being at odds with Hadamard, Einstein saw the logical character of his theory in the “correspondence” between sign relations and facts. There are thus scholars who drew a parallel between Einstein’s standpoint and Cassirer’s. For instance, Ryckman epitomises the comparison arguing that “Einstein, no more than Cassirer, is committed to apodictically valid a priori principles underlying physical knowledge”. And the addition that “the validity of general covariance as a ‘condition of possibility for natural science’ can have only a pragmatic justification stemming from its successful employment in scientific theories” (1999: 607; see also Giovanelli 2016: 147) is also in line with Cassirer’s thought. Therefore, as Winternitz’s review also shows, Einstein’s conventions parallel Cassirer’s synthetic principles since they do not originate from experience, are variable and work “as the ordering principle for words in a lexicon” (Einstein 1924a: 22, Engl. tr.).24 This reading is especially convincing with respect to the well-known letter that Einstein sent to Born on 28 June 1918: “If one just concedes to him––to Kant––the existence of synthetic judgments a priori, one is already ensnared. I must tone down (abschwächen) ‘a priori’ into ‘conventional’ in order not to have to contradict myself, but even then it does not fit in the details. Nevertheless, it is very nice reading, if not as fine as his predecessor Hume, who also had considerably more common sense” (CP 8B: doc. 575, Engl. tr.). In sum, it is arguable that Einstein opposes the neo-Kantians either when they try to “cram” GR into Kant’s system (Einstein to Schlick, October 17, 1919; CP 9, doc. 142) or when it comes to the dichotomy between transcendentalism and physical geometry.25 If this does not happen, it is unnecessary to dispute. All of this has been said bearing in mind that, according to Schlick, Einstein referred to Cassirer as a “persuader” who led people astray to induce them on the way to Kantianism (Schlick to Reichenbach, October 6, 1924, cit. in Hentschel 1990: 519).

5.4 Conclusions We started from the received view that Natorp’s and Cassirer’s interpretations of relativity theory respectively fall under immunising and revisionist strategies. Essentially, I agreed with such a vision, but I proposed some corrections especially

23 However, Friedman (1999: 41–43) explains that Schlick shifted from coordination to verification

in virtue of relativity theory and Einstein’s “rhetoric” (see also Ryckman 2005: 53 and ff.). also the review of Siegfrid Weinberg’s Erkenntnistheorie (Ferrari 1996: 146). 25 But interestingly, Ryckman (2017: 253 and ff.) has shown that Einstein’s empiricism was a “pro tem strategy” that concealed his discussion with Weyl. 24 See

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Table 5.5 Cassirer’s items outnumber Natorp’s. In general, where both are present, they are significantly in favour of Cassirer: for instance, Cassirer’s mastery of the theory is higher than that of Natorp. However, Cassirer’s immunising narrative partially undermines 10 and 1. I did not select 4 since Cassirer does not intend to quit transcendental philosophy, even though he is ready to modify it

concerning Natorp. Indeed, I introduced perspectivalist considerations according to which his reading was at least up-to-date in 1910. Furthermore, Natorp had already addressed the most important features of Cassirer’s interpretation, in particular highlighting the meaning and the role of the concept of invariance. On the other hand, I have shown that, despite his reform, Cassirer is convinced to speak on behalf of the neo-Kantian tradition. His interpretation thus creatively combines immunising with revising stances: for instance, the claim for invariance eventually leads to a more liberal concept of coordination than Natorp’s, which also accounts for the understanding of general covariance. Finally, I have explained that in Einstein the character of conventions and the spontaneity of reason reverberate originally. I have thus tried to show that his dialogue with Cassirer may have compelled him, if not to become a neo-Kantian, at least to reconsider the rejection of neo-Kantianism more carefully Ferrari (1996: 145–146). Accordingly, as an epistemologist, Cassirer proved to be more influential than Natorp, while relativity had a bearing on Cassirer’s philosophy. That being said, I provide two diagrams that summarise respectively the comparison between Natorp and Cassirer, and Cassirer’s theses (Tables 5.5 and 5.6).26

26 Hentschel (1990: 239) also provided a wide diagram with all neo-Kantian theses. It is recommendable to also read table 4 in comparison to Hentschel’s original map.

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Table 5.6 For the sake of accuracy, I have arranged revisionist and immunising claims in two levels. However, the more distinguishing character of Cassirer’s enterprise is that immunising and revising claims interweave with one another. The dashed arrow displays that there is one question left open in the book on Einstein: the connection between the ideal nature of space and time and the ontology of the field

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Chapter 6

Coordination, Geometrization, Unification: An Overview of the Reichenbach–Einstein Debate on the Unified Field Theory Program Marco Giovanelli

Abstract 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. Keywords Reichenbach · Einstein · Weyl · Unified field theory · General relativity · Geometrization · Unification · Coordination

M. Giovanelli () Università degli Studi di Torino, Department of Philosophy and Educational Sciences, Torino, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Russo Krauss, L. Laino (eds.), Philosophers and Einstein’s Relativity, Boston Studies in the Philosophy and History of Science 342, https://doi.org/10.1007/978-3-031-36498-3_6

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6.1 Introduction Most of Einstein’s published work from 1919 (Einstein, 1919b) until his death in 1955 (Einstein and Kaufman, 1955)1 is dominated by the search for a unified field theory, which aimed to unify the gravitational and electromagnetic fields into a single mathematical structure while integrating the field with its sources (Tonnelat, 1966). The history of Einstein’s engagement with such a program has been rightly described as a rapid succession of hopes and disappointments (Vizgin, 1994, 183ff.). Einstein was aware that, without an analog of the equivalence principle, the choice of the basic field structure to represent the combined electromagnetic/gravitational field could not be empirically motivated from the outset, as in the case of the metric in his theory of gravitation. Thus, in the last resort, Einstein had to rely on the criterion of mathematical simplicity, which was arbitrary to a large degree. Despite his renown for foundational work in relativity theory (Reichenbach, 1920b, 1924, 1928a), it is often overlooked that Hans Reichenbach possessed a unique combination of philosophical and technical skills that enabled him to make sense of the diverse unification efforts. Indeed, Reichenbach followed the historical development of the unified field theory-project firsthand in a way that is inextricably entangled with his personal and intellectual relationship with Einstein. In the late 1910s, Reichenbach witnessed the rise of the program when he attended the Berlin lectures and was confronted with Einstein’s skeptical reaction to Hermann Weyl’s (1918b) early attempt. In the mid-1920s, he was exposed to Einstein’s sudden change of attitude towards the unification program (Vizgin, 1994, 188). When he returned to Berlin as a professor by the end of the decade (see Hecht and Hoffmann, 1982), Reichenbach was directly involved in the journalistic craze surrounding Einstein’s latest theory (Sauer, 2006). Thereby, Reichenbach not only closely trailed the technical meanders of the field-theoretical approach to unification, but he was a privileged witness of the progressive transformation of Einstein’s philosophical outlook, from his early empiricist leanings to his later extreme rationalism. This paper aims to revisit the Einstein-Reichenbach relationship from the point of view of their evolving attitude toward the program of unifying electricity and gravitation. In particular, it considers Reichenbach not in his capacity of a staunch defender of relativity theory (Hentschel, 1982; Reichenbach, 2006), but in his less-known role of indefatigable attacker of the unified field theoryproject. Although most of Reichenbach’s critical remarks on the program appear in published writings, his private correspondence offers a more nuanced understanding of the philosophical motivations behind his mistrust towards this project. To provide

1 For

the history of the unified field theory-project, I draw freely from the standard historical literature on the subject Vizgin (1994), Goenner (2004), and Goldstein and Ritter (2003). For an overview of Einstein’s work on the unified field theory, see Sauer (2014); for the philosophical background of Einstein’s search for a unified field theory, see Dongen (2010); on Einstein’s philosophy of science, see Ryckman (2017).

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a comprehensive overview of his views, this paper focuses on three correspondences that address three different conceptual issues: Coordination: The Reichenbach-Weyl correspondence (1920–1922) In his 1920 habilitation, Reichenbach, although in passing remarks, accused Weyl of attempting to deduce physics from geometry by reducing physical reality to ‘geometrical necessity’ (Reichenbach, 1920b, 73). On the contrary, Reichenbach considered the greatest achievement of general relativity was to have shifted the question of the truth of geometry from mathematics to physics (Reichenbach, 1920b, 73). Reichenbach insisted on what he thought was the core message of Einstein’s epistemology: spacetime geometry is in itself neither true nor false, it acquires a physical meaning only when it is coordinated with the behavior of physical probes, like rods and clocks. After a correspondence with Weyl, Reichenbach (1922a, 367–368), Reichenbach accepted Weyl’s (1921d)’s defense that he did not mean to derive physics from mathematics. However, Weyl further argued that abstract spacetime geometry had nothing to do with the behavior of physical measuring devices. Reichenbach countered that, in this way, Weyl’s theory became overly formal and lost its persuasive power (Reichenbach, 1922a, 367). Geometrization: Reichenbach-Einstein correspondence (1926–1927) Reichenbach became convinced that, despite the initial failure of Weyl’s approach, Weyl’s style of doing physics was prevailing. Physicists were convinced that after the geometrization of the gravitational field, further physical insight could be obtained by geometrizing the electromagnetic field. By the end of 1922, Einstein himself started to pursue the unification program more aggressively, adopting Eddington’s approach (Einstein, 1923a). In March 1926, after making some critical remarks on Einstein’s newly published metric-affine theory (Einstein, 1925a), Reichenbach sent Einstein a 10-page ‘note’ (Reichenbach, 1926b). In it, he constructed a toy unification of the gravitational and electricity in a single geometrical framework, thereby showing that the ‘geometrization’ of a physical field was a mathematical trickery rather than a physical achievement. After a back and forth, Einstein seemed to agree (Lehmkuhl, 2014). The note was later included as section §49 in a lengthy technical Appendix to the Philosophie der Raum-Zeit-Lehre (Reichenbach, 1928a, SS46–50) in which general relativity was presented as a ‘physicalization of geometry’ rather than a ‘geometrizaton of gravitation’ (Giovanelli, 2021). Unification: Reichenbach-Einstein correspondence (1928–1929) A few months after the publication of the Philosophie der Raum-Zeit-Lehre (Reichenbach, 1928a), Einstein (1928d,b) launched yet another attempt at a unified field theory, the so-called Fernparallelismus-field theory. Reichenbach, now back in Berlin sent him once again a manuscript with some comments (Reichenbach, 1928c) and discussed the new theory in person with Einstein. This exchange of letters marked the cooling of their personal friendship but also the end of their philosophical kinship. In the late 1920s, Reichenbach (1929d,a,b) came to realize that, in Einstein’s mind, the actual goal of the unified field theory-project was not the geometrization, but the unification of two different fields. For this purpose, Einstein was willing to embrace a speculative approach to physics (Dongen,

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2010). The heuristic of mathematical simplicity gradually gained prominence in Einstein’s scientific practice, overshadowing the separation of mathematics and physics that formed the basis of the Einstein-Reichenbach philosophical alliance. The aim of this paper is not to provide new documentary material. The importance of the first episode has been recognized in Reichenbach scholarship over the past few decades (Ryckman, 1995, 1996). The other two correspondences have only recently been published and analyzed in detail (Giovanelli, 2016, 2022). However, this paper attempts to weave a coherent narrative out of these previously separate episodes, thereby shedding new light on each of them. On the one hand, this paper places the Reichenbach-Weyl debate in the broader context of Reichenbach’s negative attitude towards the unification program. On the other hand, it demonstrates how Reichenbach used the same line of argument against Einstein that he had previously used against Weyl. According to Reichenbach, the primary achievement of general relativity was the separation of mathematics and physics. Mathematics can only teach what is physically permissible but never what is physically true. Reichenbach was disappointed that some relativists had started to believe that mathematics alone could provide insights into physical reality. Thus, Reichenbach’s role as both a ‘defender’ of relativity and a ‘critic’ of further unification attempts are two sides of the same coin. Reichenbach’s disapproval of the unified field theory-project, including Einstein’s contributions to it, was also a vindication of the philosophical achievements of Einstein’s theory of gravitation: “The general theory of relativity by no means turns physics into mathematics. Quite the opposite: it brings about the recognition of a physical problem of geometry (Reichenbach, 1929c, 11). In this manner, somewhat unwittingly, Reichenbach formulated a sort of ‘theory of spacetime-theories’ (Lehmkuhl, 2017). He attempted to unravel the key to Einstein’s success in formulating a field theory of gravitation by examining the reasons for the failure of subsequent unification attempts. In doing so, Reichenbach brought to the debate, often for the first time, some of the central issues of the philosophy of space-time physics, including: (a) the relation between a theory’s abstract geometrical structures (metric, affine connection) and the behavior of physical probes (rods and clocks, free particles, etc.); (b) the question whether such an association should be regarded as a geometrization of physics or a physicalization of geometry; and (c) the interplay between geometrization and unification in the context of a field theory.

6.2 Coordination: The Weyl-Reichenbach Correspondence (1920–1921) After serving in World War I, from 1917 until 1920, Reichenbach worked in Berlin as an engineer specializing in radio technology to support himself after the death of his father. Nevertheless, in his spare time, he managed to attend Einstein’s lectures on special and general relativity in winter term 1917–1918 and in summer term 1919. We possess three sets of Reichenbach’s undated student notes (HR, 028-

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Fig. 6.1 Reichenbach’s student notes. Einstein introduces the notion of parallel transport of vectors

01-04, 028-01-03, 028-01-01). One set of notes (HR, 028-01-01) appears to be very similar to Einstein’s own lecture notes from 1919 (Einstein, 1919a). Einstein’s lectures on General Relativity were organized in a manner that closely followed the structure of his previously published presentations of relativity theory (Einstein, 1916, 1914). The mathematical apparatus of Riemannian geometry is introduced by starting from the metric .gμν as the fundamental concept, that is, from the formula to calculate the squared distance .ds 2 = gμν dxμ dxν between any two neighboring points .xν and .xν + dxν independently of the coordinate system. From the .gμν , one   μν , which enters the geodesic can calculate the so-called Christoffel symbols . τ τ equation, and the Riemann tensor .Rμνσ which generalized the Gaussian notion of curvature. However, both Reichenbach’s (see Fig. 6.1) and Einstein’s notes show that in the lectures of May-June 1919, Einstein used for the first time the interpretation of the curvature in terms of the parallel displacement of vectors, which was introduced by Tullio Levi-Civita (1916) and applied to relativity theory by Hermann Weyl (1918b). Both names are mentioned explicitly (HR, 028-01-03, 33). Instead of using the metric as a fundamental concept, it is more convenient to introduce a coordinate-independent criterion of parallelism of vectors at neighboring points .xν

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τ Aν x (HR, 028-01-03, 33).2 The . τ , which is and .xν + dxν , that is, .dAμ = μν ν μν τ =  τ ), is the so-called affine supposed to be symmetrical in the lower indexes (.μν μν connection or displacement.3 The metric could be introduced at a later stage by defining the scalar product of vectors in a way that’s independent of the coordinate system. The squared length of a vector is defined as the scalar product of the vector with itself: .l 2 = gμν Aμ Aν . By imposing the condition that the length of vectors τ are found to have remains constant under parallel transport, the coefficients of .μν   μν τ the same numerical values as the Christoffel symbols, .μν = − (up to a sign). τ In this way, the structure of Einstein-Riemann geometry can be recovered without any reference to the metric .gμν . It differs from the Euclidean structure in that, when a vector is transported along a closed curve, it acquires a rotation whose magnitude τ (g). Only when the latter vanishes can is determined by the Riemann tensor .Rμνσ vectors be considered parallel at a distance. Weyl’s technical innovation in differential geometry played a fundamental role, not only in successive presentations of general relativity (see Einstein et al., 1922, 45ff.), but more prominently in the development of the unified field theory-project. If one takes a symmetric .gμν as the fundamental variable, the Christoffel symbols are τ independently the only possible outcome. However, defining the displacement .μν   τ = − μν appears only as a of the metric .gμν , the Riemannian connection .μν τ special case that has been achieved by introducing a series of contingent conditions. Dropping some of these conditions results in additional mathematical degrees of freedom that could be used to accommodate the electromagnetic field alongside the gravitational field in a unified ‘geometrical’ description. As is well known, Weyl (1918c,a, 1919a) was bothered by a conceptual asymmetry characterizing Riemannian geometry. The comparison of direction of vectors is path-dependent, whereas the comparison of their lengths remains distant-geometrical. To compensate for this ‘mathematical injustice’ (Afriat, 2009), Weyl introduced a more general affine connection that depends not only on the gravitational tensor .gμν but also on the four-vector .ϕν , which could be identified with the electromagnetic four-potential. However, in the absence of an analogon of the equivalence principle, the justification of such identification was merely

2 Throughout

the paper, the notation used by Reichenbach (1928a) which, in turn is based on Eddington (1923, 1925) is used. 3 The affine geometry is the study of parallel lines Weyl (1918c) introduced the expression ‘affine connection’ (affiner Zusammenhang). The term ‘connection’ refers to the possibility of comparison of vectors at close points. However, it is the notion of ‘sameness’ rather than parallelism that holds significance. Thus, some authors, such as Reichenbach (1928b), prefer to use the term ‘displacement’ (Verschiebung), which emphasizes the small coordinate difference .dxν along which the vector is transferred. Note that ‘displacement’ also refers to the vector .dxν . To avoid confusion, the term ‘transfer’ Übertragung has also been used, for example, by Schouten (1922).

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formal.4 Nevertheless, Weyl came to the conclusion that his theory offered a unified geometrization of both gravitational and electromagnetic phenomena, similar to how general relativity represented a geometrization of gravitational phenomena. Weyl did not hesitate to declare that “Descartes’ dream of a purely geometrical physics” had been finally fulfilled (Weyl, 1919b, 263). Einstein had repeatedly criticized Weyl’s attempt.5 Nevertheless, after corresponding with Theodore Kaluza in the spring of 1919 (Wüensch, 2005),6 he had started to show increasing interest in the unification program (Einstein, 1919b). The question fell into the background after the success of the eclipse expedition was announced in November 1919 (Dyson et al., 1920). By the end of the year, Einstein was turned into an international celebrity, leaving him little time to work (Einstein to Fokker, Dec. 1, 1919; CPAE, Vol. 9, Doc. 187, Einstein to Hopf, Feb. 2, 1920; CPAE, Vol. 9, Doc. 295). As a trained physicist with a doctorate in philosophy (Reichenbach, 1916), Reichenbach was uniquely positioned to engage with the philosophical implications of the theory. Through his attendance at Einstein’s lectures, he had acquired a detailed technical knowledge of the new theory, surpassing that of most philosophers of his time. In February or March 1920, shortly after his move to Stuttgart, Reichenbach decided to make this topic the subject of his habilitation. According to his later recollections,7 in the preceding months, he had further worked on the theory “also according to Weyl” (HR, 04406-23)—that is, probably studying Weyl’s textbook Raum–Zeit–Materie (Weyl, 1918b). The Kapp-Pusch coup in March of 1920 provided Reichenbach with a few days off from his job at the Huth radio industry (HR, 044-06-23). This gave him the opportunity to work uninterrupted, and in just 10 days, he completed an early draft of his habilitation. The manuscript was then typed and shown to Einstein and others. Thanks to the intervention of Arnold Berliner, the influential editor of Naturwisseschaften, Reichenbach secured a publishing agreement with Springer (HR, 044-06-23). 4 In

Weyl’s (1918a) theory the vector field .ϕν determines the change of length of vectors; the curl of .ϕν is the length-curvature tensor .Fμν , which satisfy satisfy an identity which looks a lot like Maxwell-Minkowski equations in empty space. Thus, it was very suggestive to interpret .ϕν as the electromagnetic four-potential and its curl .Fμν as the electromagnetic tensor. 5 Einstein raised at least four objections against Weyl’s theory: (1) the so-called ‘measuring rod objection’ (Maßstab-Einwand) (Einstein, 1918) is most famous. Weyl’s theory predicts that the clocks’ ticking rate should depend on the clocks’ prehistory. However, the spectral lines of atoms used as clocks are well-defined; (2) the geodesic equation in Weyl’s theory contains terms proportional to the vector potential .ϕν . Thus, the electromagnetic four-vector potential affects the motion of uncharged particles; (3) the representation of the Lagrangian is the mere sum of electromagnetic and gravitational components, thus Weyl’s theory does not achieve a proper unification; (4) The field equations derived from this Lagrangian were of the fourth-order in the .gμν which, even in the absence of an electromagnetic field, did not reduce to the generally relativistic equations of gravitation, violating the correspondence principle. 6 Einstein to Kaluza, Apr. 21, 1919; CPAE, Vol. 9, Doc. 26; Einstein to Kaluza, Apr. 28, 1919; CPAE, Vol. 9, Doc. 30; Einstein to Kaluza, May 5, 1919; CPAE, Vol. 9, Doc. 35; Einstein to Kaluza, May 14, 1919; CPAE, Vol. 9, Doc. 40; Einstein to Kaluza, May 29, 1919; CPAE, Vol. 9, Doc. 48. 7 These autobiographical notes HR, 044-06-23 were written in 1927.

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6.2.1 Reichenbach’s Habilitation and His Critique of Weyl Theory For the still Kantian Reichenbach, one of the main philosophical merits of the theory of relativity was the revelation of the physical character of geometry.8 The possibility of non-Euclidean geometries had already suggested that the Euclidean character of physical space could no longer be taken for granted (Reichenbach, 1920b, 3; tr. 1969 3). According to Reichenbach, “the theory of relativity embodies an entirely new idea” (Reichenbach, 1920b, 3; tr. 1969 4). relativity theory claims that the theorems of Euclidean geometry do not apply to the physical space, that Euclidean geometry is simply false (Reichenbach, 1920b, 3; tr. 1969 4). As a result, the development of relativity theory has made it necessary to differentiate between pure geometry as a formal system with no interpretation and applied geometry as an empirical theory of physical space (Reichenbach, 1920b, 73; tr. 1969 76). The propositions of pure geometry are neither true nor false in themselves. The question of the truth of physical geometry pertains to physics alone. In order to emphasize the importance of this philosophical achievement, almost incidentally, Reichenbach indicated Weyl’s recent theory as a glaring example of how easy it was to slip into old habits. Weyl once again believed to have found a particular geometry that, for its intrinsic mathematical appeal, must have been ‘true’ for physical reality: “In this way, the old mistake is repeated” (Reichenbach, 1920b, 73; tr. 1969 76). Reichenbach’s brief outline of Weyl’s theory is sufficient to grasp the gist of his argument. As Reichenbach’s put it, “Weyl’s generalization of the theory of relativity [. . . ] abandons altogether the concept of a definite length for an infinitesimal measuring rod” (Reichenbach, 1920b, 73; tr. 1969 76). In Euclidean geometry, a vector can be shifted parallel to itself along a closed curve so that, when brought back to the point of departure, it has the same direction and the same length. In the Einstein-Riemann geometry, it has the same length, but not the same direction. In Weyl’s theory, it does not even retain the same length. As we have seen, in this way, in addition to the ‘metric tensor’ .gμν , a ‘metric vector’ .ϕν is introduced

8 Reichenbach’s

habilitation has recently attracted renewed attention (Friedman, 2001). Reichenbach borrowed from Schlick (1918) the idea that physical knowledge is, ultimately (Zuordnung), the process of relating an axiomatically defined mathematical structure to concrete empirical reality (Padovani, 2009). However, Reichenbach attempted to give this insight a ‘Kantian’ twist. According to Reichenbach, in a physical theory, besides the ‘axioms of connections’ (Verknüpfungsaxiome) encoding the mathematical structure of a theory, one needs a special class of physical principles, the ‘axioms of coordination’ (Zuordnungsaxiome), to ensure the univocal coordination of that structure to reality. For the young Reichenbach, the latter axioms are a priori because they are ‘constitutive’ of the object of a physical theory. However, they are not apodeictic or valid for all time. As is well known, Reichenbach would soon abandon the project of a constitutive but relativized a priori. However, he would firmly maintain the separation between the mathematical framework of a theory (the ‘defined side’) and the way it relates to empirical reality (the ‘undefined side’) as an essential feature of his philosophy (Reichenbach, 1920b, 40; tr. 1969 42) as an essential feature of his philosophy.

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that formally behave like the electromagnetic four-potential. Reichenbach conceded that Weyl’s theory represented a possible generalization of Einstein’s conception of spacetime that, “although not yet confirmed physically, is by no means impossible” (Reichenbach, 1920b, 76; tr. 1969 79). Reichenbach seemed to have been aware of Einstein’s main objection to Weyl’s proposal (see Einstein, 1918). In general relativity, the length ds of the time-like vector .dxν is measured by a physical clock, e.g., by the crests of waves of radiation emitted by an atom. If we maintain this interpretation, then Weyl’s theory implies that “the frequency of a clock is dependent upon its prehistory” (Reichenbach, 1920b, 77; tr. 1969 80). It particular, it is affected by the electromagnetic potentials .ϕν it has encountered. Thus, two atomic clocks, at one place, will, in general, not tick at the same rate when they are separated brought back together. This result appears to contradict a vast amount of spectroscopic data that shows that all atoms of the same type have the same systems of stripes in their characteristic spectra independently of their past history. Reichenbach conceded to Weyl that these effects might “compensate each other on the average” (Reichenbach, 1920b, 77; tr. 1969 80). Thus, the fact that “the frequency of a spectral line under otherwise equal conditions is the same on all celestial bodies” could be interpreted as an approximation, rather than a consequence of the Riemannian nature of space-time (Reichenbach, 1920b, 77; tr. 1969 81). According to Reichenbach, Weyl seems to imply that his non-Riemannian geometry must be true physically because it is mathematically superior to Riemannian geometry. As we have seen, in Weyl geometry, a vector moving around a closed loop would have the same length but a different direction, whereas in Riemannian geometry it would have different length and direction. Thus, Weyl geometry eliminated the last distant-geometrical treatment of Riemannian geometry. In this sense, Weyl geometry seems to be the most ‘general geometry’, a purely infinitesimal geometry. As a consequence, there is no reason to assume that a more special geometry applies to reality from the outset. However, Reichenbach had already surmised that this generalization could be continued. In Weyl geometry, lengths can be compared at the same point in different directions but not at distant points. “The next step in the generalization would be to assume that the vector changes its length upon turning around itself” (Reichenbach, 1920b, 76; tr. 1969 85). Probably, more complicated generalizations could be thought of. “Nothing may prevent our grandchildren from someday being confronted with a physics that has made the transition to a line element of the fourth degree” (Reichenbach, 1920b, 76; tr. 1969 79).9 Thus, there is no “‘most general’ geometry” that in and of itself must be physically true (Reichenbach, 1920b, 76; tr. 1969 80). No matter how far one pushes the level of mathematical abstraction, the “difference between physics and mathematics” (Reichenbach, 1920b, 76; tr. 1969 80) cannot be erased; geometry alone can never be sufficient to establish the reality of physical space (Reichenbach, 1920b, 76; tr. 1969 80).

9.ds 4

= gμνσ τ dxμ dxν dxσ dxτ instead of .ds 2 = gμν dxμ dxν as in Riemannian geometry.

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Reichenbach accused Weyl of neglecting the main philosophical lesson of general relativity: the unbridgeable difference between physics and mathematics. According to Reichenbach, a mathematical axiom system is indifferent to the applicability of geometry, and it “never leads to principles of an empirical theory” (Reichenbach, 1920b, 73; tr. 1969 76). On the other hand, only a physical theory can answer the question of the validity of a particular geometry for physical space (Reichenbach, 1920b, 73; tr. 1969 76). [Thus] it is incorrect to conclude, like Weyl10 and Haas,11 that mathematics and physics are but one discipline. The question concerning the validity of the axioms for the physical world must be distinguished from that concerning possible axiomatic systems. It is the merit of the theory of relativity that it has removed the question of the truth of geometry from mathematics and relegated it to physics. If now, from a general geometry, theorems are derived and asserted to be a necessary foundation of physics, the old mistake is repeated. This objection must be made to Weyl’s generalization of the theory of relativity [. . . ] Such a generalization is possible, but whether it is compatible with reality does not depend on its significance for a general local geometry. Therefore, Weyl’s generalization must be investigated from the viewpoint of a physical theory, and only experience can be used for a critical analysis. Physics is not a ‘geometrical necessity’; whoever asserts this returns to the pre-Kantian point of view where it was a necessity given by reason. (Reichenbach, 1920b, 73; tr. 1969 76)

To a certain extent, this objection contains the backbone of Reichenbach’s criticism of the unified field theory-project in the following decade. Weyl seems to have misunderstood the fundamental lesson of Einstein’s theory. The question of the “validity of axioms for the physical world” must be distinguished from that concerning “possible the axiomatic systems” (Reichenbach, 1920b, 73; tr. 1969 76). It is true that it is “a characteristic of modern physics to represent all processes in terms of mathematical equations”, and, one might add, progressively more abstract mathematics. Still, “the close connection between the two sciences must not blur their essential difference” (Reichenbach, 1920b, 33; tr. 1969 34). The truth of mathematical propositions depends upon internal relations among their terms, whereas the truth of physical propositions depends on the coordination (Zuordnung) to something external, on a connection with experience. “This distinction is due to the difference in the objects of knowledge of the two sciences” (Reichenbach, 1920b, 33; tr. 1969 34). The mathematical object of knowledge is uniquely determined by the axioms and definitions. These definitions have been called “implicit definitions” by Schlick (1918). They define one concept always through another concept without referring to external content (Reichenbach, 1920b, 33; tr. 1969 36). For this reason,

10 In

the 1919 edition of the Raum–Zeit–Materie, Weyl included a presentation of his unified field theory. Thus, the ‘Conclusion’ of the book was characterized by even more inspired rhetoric: “physics and geometry coincide with each other” (Weyl, 1919b, 263). The tendency of physicalizing geometry that prevailed among the leading protagonists of the nineteenth century from Gauss to Helmholtz seemed to be superseded by the project of geometrizing physics that ran from Clifford to Einstein: “geometry has not been physics but physics has become geometry” (Weyl, 1919b, 263). 11 Haas (1920).

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mathematics is absolutely certain and necessary. On the contrary, “the physical object cannot be determined by axioms and definitions. It is a thing of the real world, not an object of the logical world of mathematics” (Reichenbach, 1920b, 34; tr. 1969 37). For this reason, physical knowledge always implies a certain degree of approximation. As is well-known, Reichenbach abandoned the Kantian framework in which the initial uncoupling of mathematics and physics was presented. However, he never abandoned the idea that a clear-cut division of labor between mathematical necessity and physical reality was of paramount epistemological importance. This separation was the irreversible conceptual shift that relativity theory had forced upon philosophy. On June 24, 1920, Einstein praised Reichenbach’s Habilitationschrift in a letter to Schlick (Einstein to Schlick, Apr. 19, 1920; CPAE, Vol. 9, Doc. 378). A few days later, Reichenbach asked Einstein to dedicate the book to him, insisting on the philosophical significance of relativity theory: “very few among tenured philosophers have the faintest idea that your theory performed philosophical act and that your physical conceptions contain more philosophy than all the multivolume works by the epigones of the great Kant” (Reichenbach to Einstein, Jun. 13, 1920; CPAE, Vol. 10, Doc. 57). Einstein conceded that the theory might have had philosophical relevance: “The value of the th. of rel. for philosophy seems to me to be that it exposed the dubiousness of certain concepts that even in philosophy were recognized as small change [Scheidemünzen]” (Einstein to Reichenbach, Jun. 30, 1920; CPAE, Vol. 10, Doc. 66). Alleged a priori principles are like those parvenus that are ashamed of their humble origin and try to deny it: “[c]oncepts are simply empty when they stop being firmly linked to experience” (Einstein to Reichenbach, Jun. 30, 1920; CPAE, Vol. 10, Doc. 66). Einstein’s remark, which Reichenbach would later quote in his published writing (Reichenbach, 1922a, 354), sealed a sort of philosophical alliance between them. Against Weyl’s speculative style of doing physics, which reduced physical reality to geometrical necessity, Einstein defended a clear-cut separation between geometrical necessity and physical reality. However, as we shall see, this philosophical covenant would be broken less than a decade later.

6.2.2 The Reichenbach-Weyl Correspondence Reichenbach’s book was published in a timely manner a few months later in September 1920, on the occasion of the 86th Versammlung der Gesellschaft Deutscher Naturforscher und Ärzte in Bad Nauheim. This meeting was of fundamental importance in the history of relativity theory, not least because of the famous debate between Einstein and Philipp Lenard on general relativity (Dongen, 2007). At this meeting, Reichenbach met Weyl for the first time, who gave a talk on his unified theory (Weyl, 1920a). Reichenbach may have attended the debate that followed Weyl’s talk, in which Einstein rehearsed his objections against Weyl’s theory and at the same time defended the possibility of a field theory of matter against Pauli’s attacks. Einstein’s famous lecture on ‘geometry and experience’ at

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the end of January of 1921 Geometrie und Erfahrung was probably meant to address the epistemological issues that had emerged at Bad Nauheim (Giovanelli, 2014). Reichenbach sent around copies of his Relativittstheorie und Erkenntnis apriori (Reichenbach, 1920b). Schlick, who did not attend Bad Nauheim, received the book in those days. Writing to Einstein, he praised it but complained about his critique of conventionalism (Schlick to Einstein, Sep. 23, 1920). The five letters that Reichenbach exchanged with Schlick between October and November 192012 turned out to be of fundamental importance in his intellectual biography, inducing him to abandon his early Kantianism in favor of a form conventionalism with empiricist traits.13 Despite the rather severe criticisms he had expressed in the book (Rynasiewicz, 2005), Reichenbach must have sent a copy to Weyl as well. Weyl replied with some delay on February of 1921. He did not appear to be upset by Reichenbach’s objections and responded amicably to some issues that he felt “concerned less the philosophical than the physical” (Weyl to Reichenbach, Feb. 2, 1921; HR, 015-68-04). In particular, Weyl denied ever claiming that physics had been absorbed into mathematics: It is certainly not true, as you say on p. 73, that, for me, mathematics (!!, e.g. theory of the .ζ -function?) and physics are growing together into a single discipline. I have claimed only that the concepts in geometry and field physics have come to coincide [. . . ] As for my extended theory of relativity, I cannot admit that the epistemological situation is in any way different from that of Einstein. [. . . ] Experience is in no way anticipated by the assumption of that general metric; that the laws of nature, to which the propagation of action in the ether is bound, can be of such a nature that they do not allow any curvature. [. . . ] What I stand for is simply this: The integrability of length transfer (if it exists, but I don’t think so, because I don’t see the slightest dubious reason for it) does not lie in the nature of the metric medium, but can only be based on a special law of action.14 If the historical development had been different, it seems that no one would have thought of considering the Riemannian case from the outset. As far as the notorious ‘dependence on the previous history’ is concerned, I probably expressed my opinion clearly enough in Nauheim. (Weyl to Reichenbach, Feb. 2, 1921; HR, 015-68-04)

12 Schlick

to Reichenbach, Sep. 25, 1920; HR, 015-63-23 Schlick to Reichenbach, Nov. 26, 1920; HR, 015-63-22; Schlick to Reichenbach, Dec. 11, 1920; HR, 015-63-19; Reichenbach to Schlick, Nov. 29, 1920; Reichenbach to Schlick, Sep. 10, 1920. 13 Reichenbach was confronted with Schlick’s objection that his ‘axioms of coordination’ were nothing but ‘conventions’. Reichenbach initially opposed some resistance. If the coordinating principles are fully arbitrary, he feared, geometry would be empirically meaningless. In Poincaré’s conventionalism, Reichenbach missed a constraint in “the arbitrariness of the principles [. . . ], if the principles are combined” Reichenbach to Schlick, Nov. 26, 1920; HR, 015-63-22. Einstein’s famous lecture on ‘geometry and experience’ of the end January of 1921, which was published a few months lter (Einstein, 1921a), seemed to have tipped the scale in Schlick’s favor. Reichenbach (1922a) turned Einstein’s .G + P formula into his .G + F formula, where F is a ‘metric’ or universal force affecting all bodies in the same way. By setting .F = 0, geometry becomes empirically testable. Thus, Reichenbach could embrace conventionalism without accepting that the propositions of geometry are empirical meaningless. 14 That is on the field equations of the theory which, in turn, can be derived from an ‘action principle’.

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In Bad Nauheim, Weyl presented a now well-known speculative explanation for the discrepancy between the behavior of ‘ideal’ and ‘real’ rods. Essentially, Weyl suggested that the atoms used as clocks might not retain their size when transported, but rather adjust it every time to some constant field quantity, which he identified with the constant radius of the spherical curvature of every three-dimensional slice of the world (Weyl, 1920b). As a result, the geometry read off from the behavior of material bodies would appear different from the actual geometry of spacetime, due to the ‘distortion’ caused by the adjustment mechanism. In 1921, the ‘pivotal year’ for unified field theories (Vizgin, 1994, ch. 4), Weyl (followed to some extent by Eddington (1921a) and Eddington (1921b)) expanded his strategy of ‘doubling of the geometry’, including the real ‘aether geometry’ and the ‘body geometry’ distorted by the adjustment mechanism,15 in three papers intended for different audiences, published in February (Weyl, 1921f), May (Weyl, 1921c) and July (Weyl, 1921e). In the July paper, Weyl also addressed Reichenbach’s criticism publicly: From different sides,16 it has been argued against my theory that it would attempt to demonstrate in a purely speculative way something a priori about matters on which only experience can actually decide. This is a misunderstanding. Of course from the epistemological principle [aus dem erkenntnistheoretischen Prinzip] of the relativity of magnitude does not follow that the ‘tract’ displacement [Streckenübertragung] through ‘congruent displacement’ [durch kongruente Verpflanzung] is not integrable; from that principle that no fact can be derived. The principle only teaches that the integrability per se must not be retained, but, if it is realized, it must be understood as the outflow [Ausfluß] of a law of nature (Weyl, 1921b, 475; last emphasis mine)

As Weyl explains in this passage, he never claimed that his geometry entails in its mathematical structure alone the a priori justification of its physical truth. On the contrary, he questioned the supposed a priori status of the assumption that the comparison of lengths is path-independent. For this reason, Weyl did not deny the well-established empirical fact that the spectral lines of two atoms of the same chemical substance, placed identically in the same conditions, remain unaffected by their prehistory. However, he maintains that, in principle, the physical behavior of atoms does not have anything to do with the abstract notion of parallel transport of vectors.17 Einstein assumed as an empirical fact that the ratio of the wave lengths

15 A

different variation of this strategy of ‘doubling the geometry’ was suggested by Eddington (1921a) at about the same time. He considered non-Riemannian geometries as mere ‘graphical representations’ that might serve to organize different theories into a common mathematical framework. The “natural geometry” remains exactly Riemannian (Eddington, 1921a). 16 The reference is to Reichenbach (1920b) and Freundlich (1920) who, however, refers to Haas (1920). 17 In September 1921, Pauli’s (1921) encyclopedia article on relativity theory was published as part of the fifth volume of the Enzyklopädie der Mathematischen Wissenschaften. In the chapter dedicated to Weyl’s theory, Pauli suggested that Weyl provided two different versions of the theory. In its first version, Weyl’s theory sought to make predictions on the behavior of rods and clocks, just like Einstein’s theory. From this point of view, the theory is empirically meaningful, but inadequate because of the existence of atoms with sharp spectral lines. Later, Weyl renounced this interpretation. The ideal process of the congruent displacement vectors has nothing to do with

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of two spectral lines is a physical constant that can be used to normalize the ds. Weyl, on the contrary, claims that the wave lengths of two spectral lines are always a multiple of a certain field quantity of dimension of a length that can be used to normalize the ds.

6.2.3 The Weyl-Reichenbach Appeasement Weyl’s paper referencing Reichenbach appeared at the beginning of September (Weyl, 1921e). A few weeks later, Reichenbach and Weyl met again in Jena on the occasion of the first Deutsche Physiker- und Mathematikertag, the first national scientific meeting held independently from the meetings of the Gesellschaft Deutscher Naturforscher und Ärzte. Weyl gave a talk in which he tried to provide a mathematical justification for the quadratic or Pythagorean nature of the metric (Weyl, 1921a). Reichenbach presented a report of his work on the axiomatization of relativity (Reichenbach, 1921). This report is the first written testimony of the development of Reichenbach’s philosophy after the Schlick correspondence. Reichenbach suggested that, in a physical theory, one should distinguish the axioms as an empirical proposition about light rays, rods and clocks, etc. and the definitions that establish the conceptual framework of the theory (Reichenbach to Einstein, Dec. 5, 1921; CPAE, Vol. 12, Doc. 266). After the paper came out by the end of the year (Reichenbach, 1921), Reichenbach must have sent a copy to Weyl in a missing letter of January 8, 1922. He might have included a personal retraction of his criticisms. However, the letter, which is no longer extant, reached Weyl only months later (Weyl to Reichenbach, Mar. 3, 1922; HR, 015-68-03), since he was in Barcelona, where he was giving his Catalonian Lectures (Weyl, 1923). However, Reichenbach soon issued a public retraction. During that time, he was working on a lengthy review article about philosophical interpretations of relativity that he finished in the Spring of 1922. In March, Erwin Freundlich sent the proofs of the paper to Einstein (Freundlich to Einstein, Mar. 24, 1922; CPAE, Vol. 13, Doc. 109), who generally concurred with Reichenbach’s analysis (Einstein to Reichenbach, Mar. 27, 1922; CPAE, Vol. 13, Doc. 119). The paper reviewed the most significant philosophical interpretations of relativity. However, it also included a last section on Weyl’s unified field theory: “One cannot conclude an exposition of relativistic philosophy without considering the important extension that Weyl bestowed on the problem of space 3 years ago”Man darf eine Darstellung der relativistischen Philosophie nicht abschließen, ohne der wichtigen Erweiterung

the real behavior of rods and clocks (Pauli, 1921, 763; tr. 1958, 196). However, in this way, the theory furnishes only “formal, and not physical evidence for a connection between [the] world metric and electricity” (Pauli, 1921, 763; tr. 1958, 196). In this form, Pauli argues, the theory loses its “convincing power [Überzeugungskraft]” (Pauli, 1921, 763; tr. 1958, 196).

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zu gedenken, die vor 3 Jahren Weyl dem Raumproblem zuteil werden ließ (Reichenbach, 1922a, 365), Reichenbach commented. Reichenbach now appears to be fully committed to conventionalism. The choice between Euclidean and non-Euclidean geometries is based on a convention about which rods should be considered rigid (Reichenbach, 1922a, 366). This convention is arbitrary but can be fixed by postulating that metrical forces should be eliminated (Reichenbach, 1922b). However, both Euclidean and non-Euclidean geometries implicitly assume the validity of an axiom based on an empirical fact: rods of equal length in one place can be obtained in another place, regardless of the prehistory of each rod. If this were not the case, a different definition of the unit of length would have to be given for every point in space. Reichenbach referred to this tacit assumption as the “axiom of the Riemann class” (Reichenbach, 1922a, 366). Weyl’s merit is to have demonstrated that this axiom, although quite natural, is not necessary and can be challenged. From this point of view, what Weyl achieved is a purely mathematical result: “Weyl’s great discovery is that he uncovered a more general type of manifold, of which Riemann’s space is only a special case” (Reichenbach, 1922a, 365). The fact that he tried to follow this path, regardless of its empirical correctness, was a “genial advance [genialer Vorstoß]” in the philosophical foundation of the relations between geometry and physics (Reichenbach, 1922a, 367f.). Concerning the application of this mathematical apparatus to reality, Reichenbach embraces what might be called the two-theory interpretation:18 W-I In Weyl geometry, as in Riemannian geometry, the length of vectors .l 2 = gμν Aν Aμ can be compared at the same point in different directions. Weyl dropped the assumption that l remains unchanged under parallel transport at a distant point. If a vector of length l is displaced from .xν to .xν + dxν , it will, in general, have a new length .l + dl, so that .dl/ l = ϕν dxν . “The change in scale is measured by 4 quantities .ϕμ forming a vector field”. As Reichenbach pointed out, “this procedure is a purely mathematical discovery” (Reichenbach, 1922a, 366), and as such is neither true nor false. It acquires a physical meaning if one coordinates the length l as readings of some physical measuring instruments. In general relativity, the length ds of the time-like vector .dxν is measured by a clock, e.g., the spectral lines of an atomic clock. Weyl considered natural to maintain this interpretation, so that it is “still possible to measure also in this case” (Reichenbach, 1922a, 366). However, the existence of atoms with the same spectral lines shows that clocks, even in the presence of the electromagnetic field, behave differently than predicted by Weyl’s theory. Thus, it turned out that this axiom “is quite well fulfilled in reality, so the first way of generalization seems unsuitable. The latter was therefore rejected by Weyl” (Reichenbach, 1922a, 366).

18 Reichenbach

might have been inspired by Pauli (1921). However, his name is not mentioned.

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W-II Weyl adopted a different strategy. He “defines an ideal process of scale transfer, which, however, has nothing to do with the behavior of real scales” (Reichenbach, 1922a, 367). He needs this “transplantation process” only because, he “he wants to identify the vector field .ϕν with the electromagnetic potential” , like in general relativity the .gμν were identified with the gravitational potentials (Reichenbach, 1922a, 367). Once one has individuated the basic geometrical field-quantities, the next step is to find the field equations “then obvious forms for the most general physical equations arise” via “the ‘action principle’ [Wirkungsprinzip]” (Reichenbach, 1922a,  367)—a variational principle applied to the invariant integral . Wdx for a specific Lagrangian .W. According to Reichenbach, however, in this way the “theory loses its convincing character [überzeugenden Charakter] and comes dangerously close to a mathematical formalism”19 (Reichenbach, 1922a, 367). For this reason, according to Reichenbach, “Weyl’s theory is viewed very cautiously by physicists (especially by Einstein)” (Reichenbach, 1922a, 367). Ultimately, Reichenbach seems to imply that both strategies led to a dead end. From the point of view of W-I, Weyl’s infinitesimal geometry is physically inadequate; from the point of view of W-II, it is physically empty. Nevertheless, Reichenbach conceded that his objection against Weyl’s theory in his 1920 booklet missed the point. Neither W-I nor W-II can be considered attempts to prove a priori that Weyl’s non-Riemannian geometry must be true for reality because it is mathematically preferable as a truly infinitesimal geometry: However, I have to retract my earlier objection [Reichenbach, 1920a, 73] that Weyl wants to deduce physics from reason, after Weyl has cleared up this misunderstanding [Weyl, 1921b, 475]. Weyl takes issue with the fact that Einstein simply accepts the unequivocal transferability of the standards. He does not wish to dispute the Riemann-class axiom for natural standards, but only to demand that the validity of this axiom, since it is not logically necessary, should be understood as ‘a consequence of a law of nature’. I can only agree with Weyl’s demand; it is the importance of mathematics that they are. I can only agree with Weyl’s demand; it is the importance of mathematics that, in uncovering more general possibilities, it marks the special facts of experience as special and thus preserves physics from nativity [Simplizität]. Admittedly, Weyl succeeds in explaining the unambiguous transferability of natural standards only very imperfectly. But the fact that Weyl tried to go this way, regardless of the empirical correctness of his theory, remains an ingenious advance towards the philosophical foundation of physics. (Reichenbach, 1922a, 367f.)

Weyl’s point was not that the axiom of the Riemann class is necessarily false for a priori reasons, but on the contrary, that it is not a priori true as previously assumed. The fact that two measuring rods placed next to each other are of the same length regardless of their location cannot be a coincidence; it demands an explanation. Weyl’s explanation of the apparent Riemannian behavior of “through

19 This

choice of words is similar to that of Pauli, who claimed that Weyl’s theory in the second form lost his Überzeugunggkraft (Pauli, 1921, 763; tr. 1958, 196).

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the adaptation to the radius of ‘curvature of the world’ Krümmungsradius der Welt” (Reichenbach, 1922a, 368; fn. 1) only poses a problem rather than providing an answer. The problem would only be solved by developing a proper theory of matter. However, even if this theory could be provided, Reichenbach, like Einstein, found the idea of deducing the Riemannian behavior of real clocks from a theory based on the non-Riemannian behavior of geometrical lengths awkward. In this way, the “congruent transplantation [. . . ] remains physically empty” (Reichenbach, 1922a, 368; fn. 1). If the non-Riemannian congruent transplantation of vectors must be, Reichenbach argues, then the real rods should behave better in a non-Riemannian way. Thus, Reichenbach concluded that the main achievement of Weyl was mathematical and not physical. As has often happened in the past, mathematics enlarges the range of possibilities from which physicists can choose. However, this process is far from concluded with Weyl’s rather special affine connection: The philosophical significance of Weyl’s discovery is that it proved that the problem of space could not be considered concluded even with Riemann’s concept of space. If contemporary epistemology attempted to update Kant’s transcendental aesthetics by asserting that the geometry of experience must, in any case, possess a Riemannian structure, it would be refuted by Weyl’s theory. That Weyl geometry is at least possible for reality cannot be denied. Furthermore, it should not be assumed that Weyl’s theory has reached the highest level of generality. As Einstein (1921b) has demonstrated, Weyl’s requirement of the relativity of magnitude can also be satisfied without using Weyl’s measurement method. In addition, Eddington (1921a) developed a further generalization of which Weyl’s space [Raumklasse] is merely a special case. Eddington’s space [Raumklasse] is again included as a special case in a more general one discovered by Einstein (1921b). The merit of Schouten’s theory is that it provides the conditions under which a space [Raumklasse] can be considered the most general; these are very general conditions, such as differentiability and the like. Nevertheless, it must be acknowledged that there is no absolutely most general space. The history of the mathematical problem of space should instruct epistemology never to make such sweeping claims. There are no most general concepts. (Reichenbach, 1922a, 368; fn. 1)

This passage essentially repeats Reichenbach’s argument in his habilitation: there is nothing special in Weyl geometry. However, it also shows that, in the meantime, Reichenbach had closely followed the development of the unified field theory-project. He was familiar with Einstein’s (1921b) ‘conformal’ theory, in which distances can be compared only at a single point with light rays, but the comparison at distant points with transportable rods was not defined. Reichenbach was also familiar with Eddington’s (1921a) purely affine approach in which lengths go vectors cannot be compared even not at the same place in different directions. Moreover, it is quite impressive that he was even acquainted with Schouten’s (1922) recent systematic classification of connections. Thus, Reichenbach was already τ aware that, in principle, also the natural assumption of the symmetry of the .μν could be dropped. In general, relaxing the constraints on the symmetry of the connection and the relationship between the connection and the metric could open up many possibilities for incorporating the electromagnetic field into the geometrical structure of spacetime.

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In Reichenbach’s view, physicists should be completely free to choose among all these mathematical possibilities. However, mathematics alone cannot provide a criterion for choosing which possibility is realized in nature. Mathematics is the science of possibility, while physics is the science of reality. Once a choice has been made, it is essential to coordinate the chosen structure with the behavior of various idealized physical entities used as probes. Only in this way can the choice of the geometrical structure of spacetime be tested experimentally. E.g., in general relativity, the claim that in the presence of real gravitational fields, the spacetime geometry is non-flat can be confirmed or disconfirmed by rods-andclocks measurements. Weyl initially followed the same epistemological model W-I. In the presence of the electromagnetic field, rods and clocks should behave in a nonRiemannian way. However, this prediction turned out to be empirically inadequate. As a response, Weyl embraced W-II. He embraced a sort of conspiratorial distortion of all measuring instruments. However, in doing so, he deprived the geometrical setting of any empirical content. Weyl disagreed with Reichenbach’s historical reconstruction, but for reasons that reveal a completely different frame of mind. In a letter to Reichenbach, written when the latter’s review article was already in press, Weyl confessed that he actually never abandoned W-I in favor of W-II. As a matter of fact, he never adopted W-I in the first place. As he wrote to Reichenbach: “I never gave up the plan to identify rigid rods with my transplantation, because I’ve never had that plan”; on the contrary, “I was surprised when I said that physicists had interpreted that into my words” (Weyl to Reichenbach, May 20, 1922; HR, 015-68-02). The atoms that we use as clocks are physical systems like any other and do not have in principle any privileged relation with the abstract mathematical behavior of vectors. It is the theory that decides whether we should use them as reliable clocks or not. Generally, the physical interpretation of the theory’s mathematical structure in terms of the behavior of idealized physical entities, like rods and clocks, can only be provisional. Ultimately, one has to find the field equations governing that structure and require that solutions to these equations exist exhibiting the postulated behavior of rods and clocks. This reasoning applies to Einstein and Weyl’s theory: “Einstein has to show that from the dynamics of the rigid body, it follows that the rod always has the same length, measured in his ds. Similarly, I have to show that the rod has always had the same length normalized ds normalized by .R = const” (Weyl to Reichenbach, May 20, 1922; HR, 015-68-02). In both cases, the behavior of rods and clocks comes out as a byproduct of the theory.20 However, in Weyl’s theory, the Riemannian behavior of rods and clocks that came out at the end contradicts the non-Riemannian length connection on which the theory was based (see, e.g., Du Pasquier to Einstein, Dec. 13, 1921; CPAE, Vol. 12, Doc. 379). 20 In

such a theory, the unit of length would be defined as certain number of spacing between the atoms of a cubic crystal system; each of atom, in turn, consists of electrons and protons arranged according to a specific law. A specific solution to the field equations must provide information about all the details of this arrangement. Something similar can be said for the unit of time that correspond to the vibrations of an atom.

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6.3 Geometrization: The Reichenbach-Einstein Correspondence (1926–1927) Up to this point, Reichenbach had good reasons to believe that his criticisms of Weyl’s approach broadly agreed with Einstein’s point of view. Thus, Einstein continued to express skepticism towards Weyl’s ‘Hegelian’ approach to physics (Einstein to Zangger, Jan. 1, 1921; CPAE, Vol. 12, Doc. 5). He lamented a lack of “physical clues” for these attempts at unification (Einstein to Lorentz, Jun. 30, 1921; CPAE, Vol. 12, Doc. 163) that were therefore still too speculative (Einstein to Weyl, Jun. 6, 1922; CPAE, Vol. 13, Doc. 219; Einstein to Zangger, Jun. 18, 22; CPAE, Vol. 13, Doc. 241). However, the situation changed by the end of 1922, when Einstein, during a trip to Japan, realized that Eddington’s theory had potentialities that had not been fully exploited. On board the ship, he jotted down a five-page manuscript dated January 1923 from Singapore (CPAE, Vol. 13, Doc. 417). Curiously, the third, fourth, and fifth pages were written on the reverse side of Reichenbach’s Jena talk typescript (Reichenbach, 1922a). Eddington (1921a) had extended Weyl’s approach by using τ , rather than metrical quantities only the coefficients of an affine connection .μν .gμν and .ϕν , as fundamental variables. In this context, vectors’ lengths are not even comparable at the same place; thus, in Einstein’s view, the theory avoided Weyl’s inconsistency of having geometrical lengths behaving differently from real numbers. On February of 1923, Planck presented Einstein’s attempt to derive a set of field equations to the Prussian Academy of Sciences (Einstein, 1923b). After returning to Berlin, Einstein published two further papers on the same approach in May Einstein (1923b) and Einstein (1923c). In May of 1923, Reichenbach requested a copy of Einstein’s paper “on Eddington’s extension [Erweiterung]” (Einstein, 1923b) (Reichenbach to Einstein, May 2, 1923; EA, 20 080). He did not comment on Einstein’s unification attempt at this point.21 In his correspondence with Einstein, Reichenbach was concerned with the more mundane matter of finding a publisher for his work on the axiomatization

21 As

one might infer from Reichenbach’s later writings, his point of view might have been similar to that of Pauli. In a long letter to Eddington in September 1923, Pauli insisted that a good theory should start with “the definition of the field quantities used, and how these quantities can be measured” (Pauli to Eddington, Sep. 23, 1923; WPWB, Doc. 45). One of the great achievements of relativity theory was that the coefficients .gμν could be measured with rods and clocks. Pauli explained that Weyl attempted to pursue this strategy again but then abandoned this approach (Pauli to Eddington, Sep. 23, 1923; WPWB, Doc. 45). In this way, he produced what Eddington had rightly called a ‘graphical representation’ of the two fields in unified formalism, but not a ‘natural geometry’ found experimentally as in general relativity (see Eddington, 1923, 197). Similarly, τ ] cannot be measured directly” (Pauli in Einstein-Eddington new theory “[t]he quantities [.μν to Eddington, Sep. 23, 1923; WPWB, Doc. 45). The measurable quantities .gμν and .Fμν can be τ only through complicated calculations. Thus, not only we do not have a calculated from the .μν “ ‘natural geometry’ but also not a ‘natural theory’ ” (Pauli to Eddington, Sep. 23, 1923; WPWB, Doc. 45).

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of special relativity that he had just finished (Reichenbach to Einstein, Apr. 19, 1923; EA, 20 079, Reichenbach to Einstein, May 2, 1923; EA, 20 080, Einstein to Reichenbach, Jun. 9, 1923; EA, 20 081, Reichenbach to Einstein, Jul. 10, 1923; EA, 20 082). Due to a lack of funding, Reichenbach managed to publish the book only a year later, in March 1924. With the Axiomatik der relativistischen RaumZeit-Lehre Reichenbach (1924), Reichenbach’s philosophy started to assume a more recognizable contour. In particular, Reichenbach introduced, for the first time, his celebrated distinction between “conceptual definitions” used in mathematics and “coordinate definitions” used in physics, which relate the concept of a theory to a “piece [Ding] of reality” (Reichenbach, 1924, 5; tr. 1969, 8). There is little doubt that Reichenbach believed this epistemological model was Einsteinian in spirit. However, at about that time, Einstein explicitly confessed that he had changed his mind on the topic (Einstein, 1924, 1692, see Giovanelli, 2014). In particular, he denied that every individual concept of a theory should receive a measurementoperational justification (Einstein, 1924, 1691). Ultimately, only geometry and physics together could be compared with experience (Einstein, 1926, 19), a claim that seems to have a different meaning than Reichenbach had initially surmised. The Axiomatik der relativistischen Raum-Zeit-Lehre (Reichenbach, 1924) received a lukewarm reception from philosophers, who probably found the book overly technical. However, it was Weyl’s (1924) negative review that dealt a hard blow to Reichenbach and ended their previously amicable relationship. Reichenbach felt that Weyl had used his authority as a mathematician to attack his ‘empiricist’ reading of relativity (Reichenbach, 1925). What was worse, Reichenbach must have sensed that Weyl’s ‘geometrical’ reading of relativity had taken over relativistic research. Einstein’s latest works seemed to reveal that he had also fallen under its spell.22 Unsurprisingly, Reichenbach might have felt it necessary to make a case for a different interpretation of relativity theory in a more accessible form. At about the same time, he started work on a two-volume book with the ambitious title Philosophie der exakten Naturerkenntnis. Only the first volume on space and time will be published. He wrote the first chapters in March 1925 (HR, 044-06-25). During those same months, despite Max Planck’s support, Reichenbach struggled to obtain his Umhabilitation23 from Stuttgart to Berlin to be appointed to a newly created chair of natural philosophy (Hecht and Hoffmann, 1982). Reichenbach had been attacked for his pacifist positions during the war, but after the situation seemed to have turned for the better, he started working more consistently on his book project in October 1925. He interrupted the drafting of the manuscript to follow the emerging quantum revolution at the turn of 1926 and must have started again a few months later: “March-April 1926 Weyl’s theory was worked on, and the

22 As

Weyl himself ironically remarked, Einstein undertook “the same purely speculative paths which [he was] earlier always protesting against” (Weyl to Einstein, May 18, 23; CPAE, Vol. 13, Doc. 30; cf. Weyl to Seelig, May 19, 1952, cit. in Seelig, 1960, 274f.). 23 The process of obtaining the venia legendi at another university.

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peculiar solution of §49 was found. The entire Appendix was also written at that time. (Correspondence with Einstein)” (HR, 044-06-25). The correspondence with Einstein mentioned in this passage has been preserved. It reveals Reichenbach’s concerns with Einstein’s style of doing physics becoming progressively more speculative, forsaking the solid empirical foundation of his old theory of gravitation.

6.3.1 Reichenbach’s Geometrization of the Electromagnetic Field During a trip to South America in 1925, Einstein became interested in the rationalistic and realistic reading of relativity proposed by Émile La déduction relativiste (Meyerson, 1925) CPAE, Vol. 14, Doc. 455, 6; March 12 who could provide a more adequate philosophical support for the search of a unified field theory than Schlick’s or Reichenbach’s ‘positivism’ (Giovanelli, 2018). However, he also realized that the Weyl-Eddington-Schouten line had dried up (CPAE, Vol. 14, Doc. 455, 9; March 17). Returning from South America, he embraced what he τ and the .g considered a new approach. He introduced non-symmetric .μν μν to be treated as independent fields in the variation. The antisymmetric part of the .gμν was the natural candidate for representing the electromagnetic field, at least for infinitely small fields. The physical test depended, as usual, on the construction of exact regular solutions corresponding to elementary particles. The paper was published in September of 1925 with the ambitious title Philosophie der exakten Naturerkenntnis (Einstein, 1925a). However, by that time, Einstein seemed to have already lost his confidence in that approach and moved on. At the turn of the year, after working on the new quantum mechanics, Reichenbach must have read Einstein’s new paper. On March 16, 1926, Reichenbach sent a letter to Einstein in which, after discussing his academic misfortunes, he made some critical remarks (Einstein, 1925a). Reichenbach was quite skeptical of the viability of Einstein’s current style of doing physics: I have read your last work on the extended Rel. Th.24 more closely, but I still can’t get rid of a sense of artificiality that characterizes all these attempts since Weyl. The idea, in itself very τ alone, serves deep, to ground the affine connection independently of the metric on the .μν only as a calculation crutch here in order to obtain differential equations for the .gμν and the .ϕν and the modifications of the Maxwell equations which allow the electron as a solution. If it worked, it would, of course, be a great success; have you achieved something along these lines with Grommer? However, the whole thing does not have the beautiful convincing power [Ueberzeugungskraft] of the connection between gravitation and the metric based on the equivalence principle of the previous theory. (Reichenbach to Einstein, Mar. 16, 1926; CPAE, Vol. 15, Doc. 224)

24 Einstein

(1925a).

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Reichenbach’s objections are quite sensible and not dissimilar to those of professional physicists.25 In general relativity, the choice of the .gμν as fundamental variables is anchored in the principle of equivalence. The latter justified the double meaning of the .gμν , as determining the behavior of rods and clocks, as well as the gravitational field. On the contrary, Einstein’s new theory introduces the nonτ independently of the metric .g symmetric affine connection .μν μν without giving to these field variables any physical motivation. The separate variation of the metric and connection was nothing more than a ‘calculation device’ to find the desired field equations. Only in hindsight, for formal reasons, the symmetric part of the .gμν was identified with the gravitational field and antisymmetric with the electromagnetic field. In this form, the theory has little he ‘convincing power’ (Überzeugungskraft)—-the same expression that Reichenbach (1922a, 367) had used for characterizing Weyl’s theory in his second form. Reichenbach would have been ready to retract his criticism, if Einstein’s theory delivered the ‘electron’. This concession, however, barely hides his skepticism that a field theory of matter was a credible possibility. Einstein replied on March 20 that he agreed with Reichenbach’s ‘.-Kritik’: “I have absolutely lost hope of going any further using these formal ways”; “without some real new thought” he continued, “it simply does not work” (Einstein to Reichenbach, Mar. 20, 26; CPAE, Vol. 15, Doc. 230). Einstein’s reaction reflects his disillusion with the attempts with an approach based on the generalization of τ . He would have Riemannian geometry by weakening the relations by .gμν and .μν probably been less ready to embrace the implications of Reichenbach’s .-critique, the requirement that the operation of parallel displacement of vectors should receive a ‘coordinative definition’ from the outset. At any rate, Reichenbach took the opportunity of Einstein’s positive reaction, and on March 31, 1926 sent him a note (Reichenbach, 1926b; see Fig. 6.2), in which he developed the .-critique in detail (Reichenbach to Einstein, Mar. 24, 1926; CPAE, Vol. 15, Doc. 235). A point-by-point commentary of Reichenbach’s note has been provided elseτ to define where (Giovanelli, 2016). Reichenbach introduced a non-symmetric .μν an operation of displacement expressing the effect of both the gravitational and electromagnetic fields. In Riemannian geometry, the straightest lines are the shortest between two points. If the connection is non-symmetric, the straightest lines generally do not coincide with the shortest (or, more precisely, the lines of extremal length). Charged mass points of unit mass move (or their velocity four-vector is

25 In

a review of the German translation (Eddington, 1925) of Eddington’s relativity textbook (Eddington, 1923) that came out a few weeks later, Pauli (1926) expressed similar concerns. Without the equivalence principle, the entire geometrization program appeared to Pauli unjustified: “An attempt at an analogous geometrical interpretation of the electromagnetic field faces the difficulty that there is no empirical fact corresponding to the equality of heavy and inert mass, which would make such an interpretation appear ‘natural’ ” (Pauli, 1926). The solution was to avoid any connection between geometry and the behavior of rods and clocks. However, in this way, one could at most produce what Eddington called a ‘graphical representation’. According to Pauli, similar objections could be raised against Einstein’s last work (Einstein, 1925b).

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Fig. 6.2 Reichenbach’s note on the geometrization of the electromagnetic field

parallel-transported) along the straightest lines, and uncharged particles move on the straightest lines that are at the same time the shortest ones (or rather, the line of extremal length) (Reichenbach, 1926b). Einstein was not impressed (Einstein to Reichenbach, Mar. 31, 1926; CPAE, Vol. 15, Doc. 239). Thus, Reichenbach rushed to point out that Einstein had misunderstood the spirit of the typescript. As Reichenbach explained, he was working on a book on the philosophy of space and time, and thereby he “wondered what the geometrical presentation of electricity actually means” (Reichenbach to Einstein, Apr. 4, 1926; CPAE, Vol. 15, Doc. 244). Reichenbach wanted to challenge the idea that geometrizing a field is per se a useful heuristic strategy: “If one succeeds in establishing unified field equations that admit the electron as a solution,

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this would be something new.” (Reichenbach to Einstein, Apr. 4, 1926; CPAE, Vol. 15, Doc. 244). To this purpose, however, the Maxwell and Einstein field equations needed to be modified: “This is the problem on which you are working and of course also what Weyl and Eddington meant” (Reichenbach to Einstein, Apr. 4, 1926; CPAE, Vol. 15, Doc. 244). However, the geometrical representation of electricity in itself does not lead to this goal. “It can at most be an aid [Hilfsmittel] to guessing the right equations”; it might be that “what looks most simple from the standpoint of Weyl geometry also happens to be correct. But this would be only a coincidence” (Reichenbach to Einstein, Apr. 4, 1926; CPAE, Vol. 15, Doc. 244). With his theory, Reichenbach “wanted to turn against the notion that something had already been gained with the geometrical presentation of electricity” (Reichenbach to Einstein, Apr. 4, 1926; CPAE, Vol. 15, Doc. 244). In comparison with Eddington or Einstein’s last proposals, Reichenbach insisted, his approach had even the advantage “that the operation of displacement possesses a physical realization [Realisierung]” (Reichenbach to Einstein, Apr. 4, 1926; CPAE, Vol. 15, Doc. 244), namely, the velocity-vector of charged mass particles. In this way, the notion of the straightest and shortest lines is physically meaningful. For this reason, in Reichenbach’s view, his geometrization was comparable to that provided by general relativity. Nevertheless, differently from Einstein’s theory of gravitation, Reichenbach’s theory did not lead to any new physical prediction. Thus, Reichenbach concluded, a successful geometrization does not necessarily lead to a successful physical theory. Although Einstein probably continued to find the technical details of Reichenbach’s attempt questionable, his philosophical point clearly resonated with Einstein: You are completely right. It is incorrect to believe that ‘geometrization’ means something essential. It is instead a mnemonic device [Eselsbrücke] to find numerical laws. If one combines geometrical representations [Vorstellungen] with a theory, it is an inessential, private issue. What is essential in Weyl is that he subjected the formulas, beyond the invariance with respect to [coordinate] transformation, to a new condition (‘gauge invariance’).26 However, this advantage is neutralized again, since one has to go to equations of the 4. order,27 which means a significant increase of arbitrariness. (Reichenbach to Einstein, Apr. 8, 1926; CPAE, Vol. 15, Doc. 249)

Einstein seems to endorse Reichenbach’s claim that a ‘geometrization’ is not an essential achievement of general relativity. However, it is worth noticing that Einstein goes beyond Reichenbach and claims that the very notion of ‘geometry’ is τ , etc. are ultimately multi-components meaningless (Lehmkuhl, 2014). The .gμν , .μν mathematical objects characterized by their transformation properties under change

26 That is, invariance by the substitution of .g ik

with .λgik where .λ is an arbitrary smooth function of position (cf Weyl, 1918b, 468). Weyl introduced the expression ‘gauge invariance’ (Eichinvarianz) in Weyl, 1919a, 114. 27 Cf. Weyl, 1918b, 477. Einstein regarded this as one of the major shortcomings of Weyl’s theory; see Einstein to Besso, Aug. 20, 1918; CPAE, Vol. 8b, Doc. 604, Einstein to Hilbert, Jun. 9, 1919; CPAE, Vol. 9, Doc. 58.

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of coordinates. There is nothing ‘geometrical’ about those quantities. Thus, Einstein’s point is only superficially similar to that of Reichenbach. Einstein declared that the difference between geometry and the rest of mathematics was inessential. On the contrary, as we shall see, Reichenbach intended to show that the difference between geometry and physics was essential. Einstein’s argument was meant to provide support to the unified field theory-project. Against those that believed that the geometrization program could not be extended beyond the gravitational field, he could argue that geometrization has never been the issue in the first place.28 Reichenbach’s argument, on the other hand, was an attack on the belief that geometrization by itself would lead to physical results.

6.3.2 The Appendix to the Philosophie der Raum-Zeit-Lehre Strengthened by Einstein’s endorsement, Reichenbach presented the note in Stuttgart at the regional meeting of the German Physical Society on May (Reichenbach, 1926a). In the following months, he must have further work on the manuscript of his book and by the end of the year, he could write to Schlick that “[t]he first volume that deals with space and time [was] finished” (Reichenbach to Schlick, Dec. 6, 1926; SN). Reichenbach hoped to publish the book in the forthcoming Springer series ‘Schriften zur wissenschaftlichen Weltauffassung’ directed by Schlick and Philipp Frank. However, Springer rejected the book as being too long. By July, Reichenbach could announce to Schlick that he had reached a publication arrangement with de Gruyter (Reichenbach to Schlick, Jul. 2, 1927; SN). The publisher agreed to publish only the first volume under the title Philosophie der Raum-Zeit-Lehre. According to Reichenbach’s recollections, the manuscript was not changed significantly after February 1927 (HR, 044-06-25). The drafts were finished in September, and the preface was dated October 1927. The note that Einstein had sent to Einstein in the Spring of 1926 became, with few changes, the §49 of an extended Appendix dedicated to the modern development of differential geometry and the problem of the geometrical interpretation of electricity. If read with the inclusion of the Appendix,29 the Philosophie der Raum-Zeit-Lehre appears as a much more complex book. It was not only as a defense of a ‘conventionalist’ reading of the foundations of geometry, as it is usually claimed; it was at the same time an attack on the widespread interpretation of general relativity as a ‘geometrization’ of the gravitational field (Reichenbach, 1928a, 294; tr. 1958, 256). The Appendix to the Philosophie der Raum-Zeit-Lehre was simply a continuation of the line of argument that was only partially developed in the last chapter of the book. “The geometrical interpretation of gravitation”, Reichenbach wrote using an effective analogy, “is merely the visual cloak in which the factual assertion” encoded 28 Pauli’s

(1926) review of the German translation of Eddington (1925) is a typical example of this type of criticism. 29 The Appendix was not included in the English translation Reichenbach (1958).

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by the equivalence principles “is dressed” (Reichenbach, 1928a, 354; tr. [493]). The cloak might be conceived as an inextensible network of rods and clocks tailored to the body of the gravitational field. However, “[i]t would be a mistake to confuse the cloak with the body which it covers; rather, we may infer the shape of the body from the shape of the cloak which it wears. After all, only the body is the object of interest in physics” (Reichenbach, 1928a, 354; tr. [493]). The fact that a Euclidean cloak, so to speak, does not fit the body of a real gravitational field allows knowing something new about the shape of the body, that is, to make the new predictions about the behavior of free-falling mass particles, light rays, clocks, etc. Unfortunately, according to Reichenbach, recent relativistic research seemed to have confused the cloak for the body itself. “The great success, which Einstein had attained with his geometrical interpretation of gravitation” led many “to believe that similar success might be obtained from a geometrical interpretation of electricity” (Reichenbach, 1928a, 352; tr. [491]). After the physics community accepted general relativity as a theory of gravitation, the search for a suitable geometrical cloak that could cover the naked body of the electromagnetic field began. The separation of the ‘operation of displacement τ from the operation of comparison of length at a distance .g of vectors’ .μν μν gave physicists new mathematical degrees of freedoms that could be exploited to accommodate the electromagnetic field alongside the gravitational field. “However, the fundamental fact which would correspond to the principle of equivalence is lacking” (Reichenbach, 1928a, 354; tr. [493]). Thus, physicists needed to proceed through trial and error in the search for suitable geometrical-field variables. Initially, attempts were made to identify these geometrical structures with the ‘true’ the geometry of spacetime. The latter was supposed to be endowed with a more general affine structure. To give this claim empirical content, Weyl initially provided a τ in terms of the behavior of rods and “realization of the process of displacement” .μν clocks. Weyl’s project failed because rods and clocks did not behave as predicted by the theory in the presence of an electromagnetic field. Weyl found “a cloak” in which he could dress the new theory, but did not have “the body that this new cloak would fit” (Reichenbach, 1928a, 353; tr. [493]). Despite the failure of Weyl’s project, physicists did not abandon the geometrization program. Instead, they came to the conclusion that “such ‘tangible’ [handgreifliche] realizations does not lead to the desired field equations” (Reichenbach, 1928a, 371; tr. [517]). Thus, theories were proposed by “Weyl, Eddington and Einstein” that “renounced such a realization of the process of displacement” (Reichenbach, 1928a, 371; tr. [517]). The geometrical structures chosen, such τ , .ϕ , the . τ , and .g , did not have any physical meaning from as the .μν ν μν μν the outset, i.e., the values of the coefficients of those ‘structures’ were not the results of measurements. Einstein himself “has devised several new formulations in which the geometrical interpretation is reduced to the role of a mathematical tool [Rechenhilfsmittels]” (Reichenbach, 1928a, 369; tr. [516]). The key was to identify appropriate dynamical variables that could be used to construct the correct action and obtain the desired equations. However, since the fundamental variables did

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not have any physical meaning, the resulting field equations could not be directly compared with experience, as in the case of general relativity. The field equations could be confronted with reality only by integrating them in the hope that they yield a solution for the ‘electron’. On the one hand, in Maxwell’s electrodynamics the cohesion of the electron’s charge has always been attributed to a ‘foreign force’; namely, the force of cohesion that keeps the Coulomb forces from exploding. Einstein’s theory of gravitation, on the other hand, does not imply any effect of gravitation on charge and cannot, therefore, yield the cohesive force. To find a solution to the problem of matter, Maxwell’s and Einstein’s field equations should be valid to “a high degree of approximation” to recover the success of previous theories in the case of weak fields; yet they should be “changed, because, otherwise, they would never give us the electron as a solution” (Reichenbach, 1928a, 370; tr. [517]). Physicists have to guess what kind of change has to be put forward. Without an analogon of the equivalence principle, they have become convinced that, “[i]n this ‘guessing’, the geometrical interpretation of electricity is supposed to be the guide” (Reichenbach, 1928a, 371; tr. [517]). The point of departure in this approach was “the (unwritten) assumption that whatever looks simple and natural from the viewpoint of the geometrical interpretation will lead to the desired changes in the equations of the field” (Reichenbach, 1928a, 370; tr. [517]). “The many ruins along this road”, Reichenbach pointed out, should have suggested physicists “that solutions should be sought in an entirely different direction” (Reichenbach, 1928a, 370; tr. [517]). Why did they persist? Reichenbach quite perceptively grasped their psychological motivation: “It is not the geometrical interpretation of electricity” but a deeper assumption which lies at the basis of all these attempts; namely, “the assumption that the road to a simple conception, in the sense of a geometrical interpretation, is also the road to true relationships in nature” (Reichenbach, 1928a, 370; tr. [517]). The geometrical interpretation provided by general relativity was based on a physical hypothesis, the equivalence principle, which, in turn, was based on an empirical fact, the identity of gravitational and inertial mass. The unified field theory-project is based on a different physical hypothesis of a more speculative nature, the hypothesis that the world is geometrically simple. Indeed, by reading papers on the unified field theory, one is struck by the fact that they are full of expressions like ‘most natural assumption’, ‘simplest invariant’, etc. (Reichenbach, 1928a, 370; tr. [517]). Needless to say, Reichenbach considered the idea that the ‘simplicity’ of a geometrical setting could have a bearing on its physical truth to be the consequence of a severe conceptual mistake (Reichenbach, 1928a, 372; tr. [519]). He conceded that the final decision on whether the unified field theory-project is worth pursuing “must be left to the physicist, whose physical instinct provides the sole illumination” (Reichenbach, 1928a, 372; tr. [519]). Ultimately, the scientists’ “physical instinct”, their deep conviction that the world is mathematically simple, pertains to the realm of the logic of discovery and thus lies outside the competence of epistemology. However, Reichenbach made no secret of the fact that he hoped to protect scientists from “the sirens’ enchantment [Sirenenzauber] of a unified field theory” by denouncing, once again, physicists’ never-ending temptation to blur mathematics and physics (Reichenbach, 1928a, 373).

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6.4 Unification: Reichenbach-Einstein Correspondence (1929–1930) In October 1927, Reichenbach moved back to Berlin, where he took the position of an “unofficial associate professor” (Hecht and Hoffmann, 1982). At about the same time, Einstein read the manuscript of the Philosophie der Raum-Zeit-Lehre (Einstein to Elsa Einstein, Oct. 23, 1927; CPAE, Vol. 16, Doc. 34). Soon after that, he wrote a short book review. Einstein was quite perceptive noting the two themes that Reichenbach had treated in the Appendix: (1) “In the Appendix, the foundation of the Weyl-Eddington theory is treated in a clear way and in particular the delicate question of the coordination of these theories to reality” (Einstein, 1928c, 20; m.e.). As we have seen, Reichenbach had insisted that, as in any other theory, also in unified field theory based on the affine connection is a fundamental variable, one should give physical meaning to the operation of displacement from the outset. Einstein did not comment further on this issue since he realized that this requirement was too strict. However, Einstein seemed to fully agree with the second point made by Reichenbach: (2) In the Appendix, “in my opinion quite rightly—it is argued that the claim that general relativity is an attempt to reduce physics to geometry is unfounded” (Einstein, 1928c, 20; m.e.). As we have seen, Reichenbach and Einstein had already discussed this topic in a private correspondence less than 2 years earlier (Sect. 6.3). At about the same time, Einstein published a more extensive review of Meyerson’s La déduction relativiste (Meyerson, 1925) in the Revue philosophique de la France et de l’étranger. The review reveals how Einstein’s perspective had become quite different from that of Reichenbach on both issues. Einstein embraced Meyerson’s rationalist philosophy, insisting on the deductive-speculative nature of physics’ enterprise, implicitly disavowing the operational-empirical rhetoric that seemed to have dominated his early philosophical pronouncements. However, Einstein strongly disagreed with Meyerson’s insistence that Weyl’s and Eddington’s theories were the crowning moment of a long process of geometrization of physics. He insisted again that geometry in this context is “devoid of meaning” (Einstein, 1928a, 165; m.e.). He clarified, however, that the essential point of the theories of Weyl and Eddington was not to geometrize the electromagnetic field, but to “represent gravitation and electromagnetic under a unified point of view, whereas beforehand these fields entered the theory as logically independent structures” (Einstein, 1928a, 165; m.e.). Einstein’s further attempts at unified field theory in the immediately following months reveal more clearly the reasons behind his philosophical turnabout Giovanelli (2018). During a period of illness in the spring of 1928, Einstein came up with a new proposal for a unified field theory. On June 7, 1928 Planck presented to the Prussian Academy a note on a ‘Riemannian Geometry, Maintaining the Concept of Distant Parallelism’ (Einstein, 1928d)—a flat space-time that is nonetheless τ is non-symmetrical. He introduced a new non-Euclidean since the connection .μν formalism based on the concept of n-Bein (or n-legs), n unit orthogonal vectors

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Fig. 6.3 First page of Reichenbach’s manuscript (Reichenbach, 1928c)

representing a local coordinate system attached to a point of an n dimensional continuum. Vectors at distant points are considered as equal and parallel if they have the same local coordinates with respect to their n-bein. The vierbein-field .hνa defines both the metric tensor .gμν and the electromagnetic four-potential .ϕμ . Its sixteen components can be considered as the fundamental dynamical variables of the theory. The question arises as to the field equations that determine the vierbeinfield. In his second paper, submitted on June 14, Einstein derived the field equations for the vierbein-field using a variational principle (Einstein, 1928b). A few months later, after the paper had been published, Reichenbach was able to prepare a typewritten note (Reichenbach, 1928b; see Fig. 6.3) containing some

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comments which he sent to Einstein for feedback: Dear Herr Einstein, I did some serious thinking on your work on the field theory and I found that the geometrical construction can be presented better in a different form. I send you the ms. enclosed. Concerning the physical application of your work, frankly speaking, it did not convince me much. If geometrical interpretation must be, then I found my approach simply more beautiful, in which the straightest line at least means something. Or do you have further expectations for your new work? (Reichenbach to Einstein, Oct. 17, 1928; EA, 20-92; m.e.)

In this passage, Reichenbach makes two unrelated points, which, however, appear to form a single two-pronged argument. In the note he sent to Einstein. Reichenbach (1928c) demonstrated that—if one lets aside from the n-bein formalism—Einstein’s new geometrical settings could be easily inserted into the Weyl–Eddington–Schouten lineage, as a special case of metric space in which the connection is flat, but non-symmetric.30 If so, Reichenbach could raise the same objection he had raised against Einstein’s previous theories. According to Reichenbach, a “real physical achievement is obtained only if, moreover, the operation of displacement is filled with physical content” (Reichenbach, 1928c, 7). Einstein’s geometry, being flat, implies the existence of a straight line, a line of which all elements are parallel to each other, which is nevertheless not identical with a geodesic (Einstein, 1928b, 224). However, as Reichenbach reported, the latter has no physical meaning in Einstein’s theory. Reichenbach concluded that as a ‘geometrical interpretation’ his §49-theory was preferable since the straightest lines and shortest line correspond to the motion of charged and uncharged test particles under the influence of the combined gravitational-electromagnetic field. Once again, Einstein’s goal was to use this geometrical apparatus as a starting point to find the right ‘action’, from which a set of field equations could be derived. However, Reichenbach commented, nothing new came out of it: “[T]he derivation of the Maxwellian and gravitational equation from a variational principle was already achieved by other approaches” (Reichenbach, 1928c, 6), like, say, Einstein– Eddington purely affine theory. In the subsequent letter, Einstein defended his classification of geometries but did not comment on Reichenbach’s objection. However, he invited Reichenbach and his first wife, Elisabeth, for a cup of tea on November 5, 1928. During this meeting, Einstein might have informed Reichenbach about abandoning the variational strategy to find the field equations (Sauer, 2006). It is also probable that Einstein explained to Reichenbach that his goal was not to provide a geometrization of the electromagnetic field, but to provide their unification of both fields. Therefore, Einstein’s choice of field structure was not motivated by geometrical considerations, nor did it have a geometrical meaning. The goal was to recover a set of field 30 Starting from a general non-symmetric affine connection . τ μν

and imposing the condition that the length of vectors does not change under parallel transport, one can obtain Einstein and Riemann spaces through the “exchangeability of the specializations” (Reichenbach, 1928b, 5). By imposing that the Riemann tensor vanishes, one obtains Einstein space, while imposing that the connection is symmetric yields Riemannian space.

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equations that yielded the classical equations of gravitation and electromagnetism only to the first order. In other words, the theory should predict new effects in the case of strong fields. To obtain this result, Einstein was ready to adopt a whateverit-takes strategy. He was not only willing to forgo any physical interpretation of the fundamental variables of the theory, but he was also ready to abandon the variational approach as he was doing in the paper he was working on. It is hard to imagine that the divergence of their philosophical views did not emerge during those discussions. In a semi-popular paper Einstein had submitted a few weeks later (Einstein, 1929c, 131). Einstein insisted on the speculative nature of the new theory. One starts from this mathematical structure and then searches for the simplest and most natural field equations that the vierbein-field can satisfy (Einstein, 1929c, 131). The physical soundness of the field equations can be confirmed only by integrating them, finding particle solutions, and the laws governing their motions in the field. However, this was a challenging task. Einstein warned his readers of the dangers of proceeding “along this speculative road” (Einstein, 1929c, 127). “Meyerson’s comparison with Hegel’s program [Zielsetzung]”, Einstein put it in a footnote, “illuminates clearly the danger that one here has to fear” (Einstein, 1929c, 127).

6.4.1 Reichenbach’s Articles on Fernparallelismus Field Theory In the late 1920s, Reichenbach was a regular contributor to the Vossische Zeitung, at that time Germany’s most prestigious newspaper; not surprisingly, he was asked for a comment on Einstein’s theory, which had started attracting irrational attention in the daily press (see Pais, 1982, 346). With the advantage of having personally discussed the topic with Einstein a few weeks earlier, Reichenbach published a brief didactic paper on Einstein’s theory on January 25, 1929 (Reichenbach, 1929b). Reichenbach profited from the conversation with Einstein. In particular, it is revealing that Reichenbach did not present Einstein Fernparallelismus—as it would be more natural in popular writing—as an attempt to geometrize the electromagnetic field on par with the previous geometrization gravitational field achieved by general relativity. On the contrary, he decided to present Fernparallelismus as an attempt to unify the two separate fields on par with similar unifications operated by special and general relativity. Although the article was fairly innocuous, Einstein was distraught by Reichenbach’s decision to leak a private conversation to the press (Einstein to Vossische Zeitung, Jan. 25, 1929; EA, 73-229). The exchange of the letters that ensued (Reichenbach to Einstein, Jan. 27, 1929; CPAE, Vol. 16, Doc. 384, Einstein to Reichenbach, Jan. 30, 1929; CPAE, Vol. 16, Doc. 390, Reichenbach to Einstein, Jan. 31, 1929; CPAE, Vol. 16, Doc. 391) put a serious strain in their personal relationships. The disagreement over a seemingly trivial issue appeared to have

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been accompanied by a feeling of greater intellectual estrangement. Reichenbach’s personal correspondence expressed his disappointment with Einstein’s breach of their friendship, while his published works reflected his disappointment with Einstein’s deviation from their shared philosophical principles. By the time of the article’s publication for the Vossische Zeitung, Reichenbach had already written two papers on the Fernparallelismus that are both dated January 22, 1929 that were published in the following months (Reichenbach, 1929a,d). These articles are Reichenbach’s last contribution to issues related to relativity theory and spacetime theories. On the one hand, Reichenbach applied to Einstein’s new theory his ideas about the unified field theory-project as the were presented in the Appendix to the Philosophie der Raum-Zeit-Lehre (Reichenbach, 1928a, §46). On the other hand, he introduced novel insights by explicitly distinguishing between the ‘geometrization program’ and the ‘unification program’ In the first paper for the Zeitschrift für Angewandte Chemie, Reichenbach introduced the history of the unified field theory in an entirely different manner than he had done before. In the Appendix to the Philosophie der Raum-ZeitLehre, the history of the unified field theory program was ultimately presented as a linear evolution of the geometrization program that had progressively become more abstract. Now Reichenbach—probably following the discussion he had with Einstein in November—describes the history of the unified field theory as the progressive decline of the geometrization program and the concurrent ascent of the unification one. After the failure of Weyl’s first attempts, most physicists, including Einstein (February 1923d, May 31, 1925a), considered it preferable to sacrifice the geometrical interpretation—i.e., to relinquish the coordination of geometrical notion of parallel transport of vectors with the behavior rods and clocks—and then τ , .ϕ and so on) as ‘calculation device’ for the to use the geometrical variables (.μν ν greater good of finding the field equations. Reichenbach had come to understand that, in Einstein’s view, the aim of the unified field theory-project was not the geometrization of the electromagnetic field alongside the gravitational field; it was the unification of the electromagnetic and gravitational fields. Thus, Reichenbach’s concern became to explain what ‘unification’ means in this context. The problem was addressed in detail in the more technical paper, which grew out of the manuscript that Reichenbach had sent to Einstein (Reichenbach, 1929d). In this setting, his §49-theory came in handy. Reichenbach’s theory uses a similar geometrical setting as Einstein’s theory. Both use a non-symmetric affine connection. In Einstein’s approach, the further conditions that the geometry is flat are imposed, allowing for distant parallelism. According to Reichenbach, his §49theory was able to provide a proper geometrical interpretation of the combined gravitational/electromagnetic field. However, the theory could achieve only a formal unification because no new testable predictions were made: The author [Reichenbach] has shown that the first way can be realized in the sense of a combination of gravitation and electricity to one field, which determines the geometry of an extended Riemannian space; it is remarkable that thereby the operation of displacement receives an immediate geometrical interpretation, via the law of motion of electrically charged mass-points. The straightest line is identified with the path of electrically charged mass points, whereas the shortest line remains those of uncharged mass points. In this way,

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one achieves a certain parallelism to Einstein’s equivalence principle. By the way, [the theory introduces] a space cognate to the one used by Einstein, i.e., a metric space with τ . The aim was to show that the geometrical interpretation of electricity non-symmetrical .μν does not mean a physical value of knowledge per se. (Reichenbach, 1929d, 688; m.e.)

Suppose one wants to give a geometrical interpretation of a combined gravτ as a fundamental itational/electromagnetic field using the affine connection .μν variable. In that case, one should provide a coordinate definition of the operation of parallel displacement of vectors before starting to search for the field equations. Otherwise, it is hard to understand how one could test whether the latter made correct predictions. Reichenbach’s theory was meant to show that a successful geometrical interpretation of this kind can always be achieved with some mathematical trickery. However, more than a successful geometrization is required to achieve a substantive unification. For Reichenbach, this should have been a warning that the very hope that the geometrical interpretation of a physical field was the key to new physical insights was misplaced. Einstein Fernparallelismus-field theory is an instance of a second approach, which claims to achieve an inductive unification, by forgoing the geometrical τ in terms interpretation, that is, without providing a physical meaning of the .μν of the motion of test particles: On the contrary, Einstein’s approach of course uses the second way, since it is a matter of increasing physical knowledge; it is the goal of Einstein’s new theory to find such a concatenation of gravitation and electricity, that only in first approximation it is split in the different equations of the present theory, while is in higher approximation reveals a reciprocal influence of both fields, which could possibly lead to the understanding of unsolved questions, like the quantum puzzle. However, it seems that this goal can be achieved only if one dispenses with an immediate interpretation of the displacement, and even of the field quantities themselves. From a geometrical point of view this approach looks very unsatisfying. Its justification lies only on the fact that the above mentioned concatenation implies more physical facts that those that were needed to establish it. (Reichenbach, 1929d, 688; m.e.)

In Reichenbach’s view, Fernparallelismus appeared not only as a formally satisfying unification but also as a genuine advance over the available theories. It entails some coupling between the electromagnetic and gravitational fields that was not present in the given individual field theories. However, Reichenbach argues that Einstein could only achieve this result at the expense of a physical interpretation of the fundamental geometrical variables. As we have seen, Einstein’s τ defines a set of straight lines as privileged paths; however, flat affine connection .μν these lines are not interpreted as paths of particles (Einstein, 1930e, 23). Before integrating the field equations, the laws governing the latter are unknown (Einstein to Cartan, Jan. 7, 1930; Debever, 1979, A-XVI). Consequently, the theory cannot be confirmed or disproved experimentally by observing the behavior of suitable indicators. In Reichenbach’s diagnoses, the stagnation of the unified field theory-project depended on the presence of a sort of trade-off between geometrization and unification of which physicists were only partially aware. General relativity was

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the only theory that was able to combine both virtues: (1) the theory provided a proper geometrical interpretation of the gravitational field because it introduced a coordinative definition of the field variables .gμν , in terms of the behavior of those that were traditionally considered geometrical measuring instruments, such as rods and clocks, light rays, free-falling particles (2) the theory provided a proper unification by predicting that the gravitational field had specific effects on such measuring instruments that were not implied by previous theories of gravitation— such as gravitational time dilation (Reichenbach, 1928a, 378). Successive attempts to include the electromagnetic field in the framework of general relativistic field theory failed to uphold this standard. According to Reichenbach, the reason for this failure was ultimately not hard to pinpoint. The effective interplay between geometrization and unification did not seem reproducible without a proper analogon of equivalence principle. Without the equivalence principle, a further geometrization of electromagnetic fields was not worth pursuing since it had no physical justification.31 Einstein could counter these objections by claiming that geometrization had never been the goal. The achievement general relativity was to have combined inertial and gravitational just like special relativity has combined magnetic and electric field as components of a unified field structure. However, without an analogon of the equivalence principle, there seems to be also no physical justification for searching for further unification of the electromagnetic and gravitational field. Nevertheless, Einstein considered the separation between the two fields as theoretically unbearable (Einstein, 1930e, 24). However, he did not have any physical clue as to what the more comprehensive mathematical structure may be, in which the electromagnetic and gravitational fields will appear as two sides of the same field. Hence, Einstein had no choice but to turn to the criterion of mathematical simplicity, which was challenging to define with precision. To Reichenbach’s dismay, Einstein had abandoned the physical heuristic32 that leads him to general relativity in the name of a mathematical heuristic that was not different from Weyl’s speculative approach that he had dismissed a decade earlier. As we have seen, as early as in his habilitation, he considered the great achievement of relativity theory the separation of mathematical necessity and physical reality. Reichenbach had always perceived this separation as nothing more than a philosophical distillation of Einstein’s scientific practice. However, in the search for a unified field theory, Einstein had come implicitly to question this distinction, coming close to a plea for a reduction of physical reality to mathematical necessity. Einstein put it candidly in his Stodola-Festschrift’s contribution—that he sent for publication toward the end of January (Einstein to Honegger, Jan. 30, 1929; CPAE, abs. 864). The ultimate goal of understanding reality is achieved when one could prove that “even God could not have established these connections otherwise than they actually are, just as little as it would have been in his power to make the number 4 a prime number” (Einstein, 1929c, 127). 31 See 32 For

Footnote 17. Einstein’s earlier ‘logic of discovery’, see Giovanelli (2020).

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6.5 Conclusion After the publication of the new derivation of the Fernparallelismus-field equations in January 1929 (Einstein, 1929d), Einstein wrote a popular account of the theory for the New York Times and The Times of London (Einstein, 1929a,b, also published as Einstein, 1930d). In this article, Einstein emphasized the highly speculative nature of unified field theory-project, without hesitating to endorse Meyerson’s somewhat outrageous comparison with Hegel. It is difficult to deny the symbolic significance of Einstein’s decision to mention Meyerson rather than Reichenbach as a philosophical interlocutor in an article with such a vast readership. After a decade of personal friendship and intellectual collaboration, Einstein appears to have questioned the very core of his early philosophical alliance with Reichenbach. While Reichenbach considered the separation between mathematics and physics the great achievement of the relativity theory, Einstein regarded mathematics itself as the key to accessing the structure of the ‘total field’. Although Einstein’s Fernparallelismus attracted the attention of mathematicians, Reichenbach’s skepticism was shared by the physics community.33 Einstein was well aware of the marginality of his position, but throughout 1929, he continued to express his confidence in the Fernparallelismus-program. He defended the theory in public talks (Einstein, 1930c,b,a) as well as in private correspondence (Pauli to Einstein, Dec. 19, 1929; WPWB, Doc. 239; Einstein to Pauli, Dec. 19, 1929; WPWB, Doc. 140). However, only a few months later, Einstein and Walther Mayer presented a new approach (Einstein and Mayer, 1931), which generalized the nbein formalism to five dimensions. The optimism faded quickly again, as the theory was unable to solve the matter problem. In a popular talk given in Vienna around mid-October 1931, Einstein resigned himself to describing his field-theoretical work since general relativity as a “cemetery of buried hopes” (Einstein, 1932, 441).34 However, Einstein’s philosophical motivation for continuing on this path has not changed. Many of his former philosophical allies considered this attitude hard to fathom (Frank, 1947, 215f.). However, Einstein’s 1933 Oxford lecture address 33 Weyl,

whom Einstein had always scolded for his speculative style of doing physics, could relaunch the accusation in a paper (Weyl, 1929) in which he had uncovered the gauge symmetry of the Dirac theory of the electron (Dirac, 1928a,b). “The hour of your revenge has come”, Pauli wrote to Weyl in August: “Einstein has dropped the ball of distant parallelism, which is also pure mathematics and has nothing to do with physics and you can scold him” (Pauli to Weyl, Aug. 26, 1929; WPWB, Doc. 235). As Pauli complained, writing to Einstein’s close friend Paul Ehrenfest, “God seems to have left Einstein entirely!” (Pauli to Ehrenfest, Sep. 29, 1929; WPWB, Doc. 237). 34 It is interesting to notice that one of the reasons that induced Einstein to abandon the theory was not dissimilar to Reichenbach’s criticism: “The main reason for the uselessness of the distant parallelism construction lies, I feel, in that one can attribute absolutely no physical meaning to the ‘straight lines’ of the theory, while the physically meaningful (macroscopic) equations of motion cannot be obtained from it. In other words, the .hsv give rise to no useful representation of the electromagnetic field” (Einstein to Cartan, May 21, 1932; Debever, 1979, A XXXV). Thus, for Einstein, it was legitimate to abandon the physical interpretation of straight lines from the outset if the theory provided a way to derive the laws of motion of the electrons.

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leaves no room for doubt. Einstein’s quest for unification, he insisted, was motivated by the deep-seated conviction that “nature is the realization of the most simple mathematical ideas” (Einstein, 1933a). Einstein conceded that experience remains the sole criterion of the physical adequateness of a mathematical construction. However, he insisted that the true creative role belongs to mathematics: “I hold it to be true that pure thought is competent to comprehend the real, as the ancients dreamed” (Einstein, 1933a, 167). After all, he now claims, the search for field theories has always followed the same heuristic pattern: “the theorist’s hope of grasping the real in all its depth” lies “in the limited number of the mathematically existent simple field types, and the simple equations possible between them” (Einstein, 1933a, 168). Maxwell’s equations are the simplest laws for an antisymmetric tensor field derived from a vector; Einstein’s equations are the simplest equations for the metric tensor, etc. This strategy applies to Einstein’s last attempt at a unified field theory on a theory based on semi-vectors (Einstein and Mayer, 1932, 1933a, 1934, 1933b). After ordinary vectors, the latter are the simplest mathematical fields that are possible in four dimensions and seem to describe certain elementary particles’ properties (Dongen, 2004). One has to search for the simplest laws these semivectors satisfy (Einstein, 1933a, 168). In September 1933, 3 months after the Oxford lecture, Einstein left Europe for Princeton. Reichenbach started to teach at the University of Istanbul in the Fall of the same year. He tried to obtain a position at Princeton a few years later (Verhaegh, 2020). However, Reichenbach was concerned about Weyl’s possible opposition: “He is my adversary since a long time,” he wrote to Charles W. Morris: a supporter of a form a “mathematical mysticism” that was “very much opposed to my empiricist interpretation of relativity” (Reichenbach to Morris, Apr. 12, 1936; HR, 013-5078). Thus, in April 1936, Reichenbach turned to Einstein to ask for his support. “More than 10 years ago”, he explained, “Herr Weyl spoke out very negatively about my work on the theory of relativity”. Reichenbach feared that “Weyl’s opposition persists to these days” (Reichenbach to Einstein, Apr. 12, 1936; EA, 20-107). Reichenbach might have had good reasons for turning to Einstein’s help against Weyl in academic matters. However, it is worth noticing that, by that time, Reichenbach might have been closer to Weyl than to Einstein in scientific matters. A decade later, the roles were reversed. In the late 1930s, Weyl, like Reichenbach, had utterly lost confidence in the “geometrical leap [Luftsprünge]” of the early 1920s,35 and felt the need to “return to the solid ground of physical facts” (Weyl, 1931, 343), to the vast amount of experimental data provided by spectroscopy. On the contrary, gravitational research had turned Einstein into a ‘believing rationalist’ (Ryckman, 2014), convinced that physical truth lies in mathematical simplicity (Einstein to Lanczos, Jan. 24, 1938; EA, 15-268).

35 In

a way not dissimilar to Reichenbach, Weyl considered early unified field theories as “merely geometrical dressings (geometrische Einkleidungen) rather than as proper geometrical theories of electricity” (Weyl, 1931, 343).

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In 1938, Reichenbach finally managed to move to the United States (Verhaegh, 2020). Soon after, he and Einstein resumed their epistolary contact to support Bertrand Russell, who had been dismissed from the City College of New York due to his anti-religious stance (Reichenbach to Einstein, Aug. 14, 1940; EA, 20-127; Einstein to Reichenbach, Aug. 22, 1940; EA, 20-110). Later, both Reichenbach and Einstein contributed to a volume honoring Russell for the series Library of Living Philosophers, edited by Paul Schilpp (1944). Reichenbach was also asked to contribute to a similar volume in honor of Einstein a few years later (Schilpp, 1949). In some unpublished notes about Reichenbach’s (1949) contribution, Einstein (1949b) praised him his rare ability for combing breath of knowledge with clarity (Einstein, 1949b). However, Einstein ultimately disagreed with many of Reichenbach’s philosophical tenets. In particular, Reichenbach’s claim that “ ‘the meaning of a statement is reducible to its verifiability’36 ” appeared to Einstein problematic; he found “dubious whether this conception of ‘meaning’37 can be applied to the single statement38 ” (Einstein, 1949b). As is well known, in the so-called ‘Reply to Criticisms’ (Einstein, 1949a) included in the Schilpp-volume, Einstein reformulated this line of argument by staging a dialogue between Reichenbach-Helmholtz, Poincaré, and an anonymous non-positivists, who claims that geometry and physics can be compared with experience only as a whole (Einstein, 1949a, 676f.). The question at stake, as Einstein put it jokingly, was nothing but Pilates’s famous question ‘What is truth?’ (John 18:38, quoted in Einstein, 1949a, 676). Although this dialogue has become enormously famous, its meaning has been ultimately misunderstood. Einstein was not engaging in a philosophical digression about the nineteenth century debate on the foundation of geometry. The question what it means for a theory to be ‘true’ was ultimately motivated by his tireless pursuit of the theory of the ‘total field’. At that time, Einstein had returned to his 1925 metric-affine approach introducing τ as fundamental variables (Einstein, 1945; Einstein and non-symmetric .gμν and .μν Straus, 1945). In private correspondence, Einstein’s long-life friend Michele Besso raised against Einstein objections similar to those that Reichenbach had advanced over 20 years earlier against the same theory. The symmetric part of the .gμν and the τ , Besso claimed, should define the straightest line, which is also corresponding .μν the shortest. Do these lines represent the trajectories of test particles? What is their physical meaning? (Besso to Einstein, Apr. 11, 1950; Speziali, 1972, Doc. 171). Einstein’s reply reveals his fundamental philosophical conundrum: Your questions are entirely legitimate, but it is not answerable for the time being [. . . ]. This is because there is no real definition of the field in a consistent field theory. This puts you in a Don Quixotic situation, in that you have absolutely no guarantee whether it is ever possible to know if the theory is ‘true’. A priori there is no bridge to empiricism. For example, there isn’t a ‘particle’ in the strict sense of the word because the existence of particles doesn’t

36 In

English in the text. English in the text. 38 In English in the text. 37 In

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fit the program of representing reality by everywhere continuous [. . . ]. For example, the theory introduces a symmetric .gμν [. . . ] and then a geodesic line. However, from the outset, one has no clue that these lines have any physical meaning, not even approximately [. . . ] It boils down to the fact that a comparison with what is empirically known can only be expected from the fact that strict solutions of the system of equations can be expected found, that reproduce the behavior of empirically ‘known’ structures and their interactions. Since this is extremely difficult, contemporary physicists’ skeptical attitude is probably entirely understandable. In order to really grasp this conviction of mine, you must read my answer in the anthology [Sammelband]39 again and again. (Einstein to Besso, Apr. 15, 1950; Speziali, 1972, Doc. 172)

This passage summarizes many of the issues that Reichenbach and Einstein had discussed over the years. It explains Reichenbach’s legitimate concern that geometrical concepts of the theory, like that of the straightest lines, should receive a physical interpretation from the outset in terms of the motion of test particles. However, it also explains why Einstein did not find this approach viable in pursuing a theory in which there are stricto sensu no particles. Usually, a field is defined in the first place by the forces that it exerts on test particles. However, discovering this force law requires the integration of the field equations. It is in this context that question of the ‘truth’ of a theory of this kind could not be avoided. It was ultimately this question that Reichenbach and Einstein had discussed for over 30 years. Whereas Einstein was ready to change his conception of ‘truth’ for the search of the unified field theory, Reichenbach urged Einstein to abandon this search in the name of a once shared conception of the ‘truth’ of a physical theory.

Abbreviations CPAE EA HR SN WPWB

Einstein, Albert. 1987–. The Collected Papers of Albert Einstein. Edited by John Stachel et al. 15 vols. Princeton: Princeton University Press. The Albert Einstein Archives at the Hebrew University of Jerusalem. Archives of Scientific Philosophy: The Hans Reichenbach Papers. 1891–1953. Schlick Nachlass. Noord-Hollands Archief, Haarlem. Pauli, Wolfgang. 1979–. Wissenschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg u.a. Edited by Karl von Meyenn. 4 vols. Berlin/Heidelberg: Springer.

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39 Einstein

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——. 2006. Defending Einstein: Hans Reichenbach’s Writings on Space, Time, and Motion, ed. Steven Gimbel and Anke Walz. Cambridge: Cambridge University Press. Ryckman, Thomas. 1995. Weyl, Reichenbach and the Epistemology of Geometry. Studies in History and Philosophy of Science 25:831–870. ——. 1996. Einstein Agonists: Weyl and Reichenbach on Geometry and the General Theory of Relativity. In Origins of Logical Empiricism, ed. Ronald N. Giere, 165–209. Minneapolis: University of Minnesota Press. ——. 2014. ‘A Believing Rationalist’: Einstein and the ‘The Truly Valuable’ in Kant. In Janssen and Lehner, 377–397. ——. 2017. Einstein. New York: Routledge. Rynasiewicz, Robert. 2005. Weyl vs. Reichenbach on Lichtgeometrie. In Kox and Eisenstaedt, 2005, 137–156. Boston: Birkhäuser. Sauer, Tilman. 2006. Field Equations in Teleparallel Space-time: Einstein’s Fernparallelismus Approach toward Unified field Theory. Historia Mathematica 33:399–439. ——. 2014. Einstein’s Unified Field Theory Program. In Janssen and Lehner, 2014, 281–305. Schilpp, Paul Arthur, ed. 1944. The Philosophy of Bertrand Russell. Evanston: The Library of Living Philosophers. ——, ed. 1949. Albert Einstein: Philosopher-Scientist. Evanston, M.: The Library of Living Philosophers. Schlick, Moritz. 1918. Allgemeine Erkenntnisslehre. Berlin: Springer. Repr. in Schlick, 2006, Vol. 1. ——. 2006. Gesamtausgabe, ed. Friedrich Stadler and Hans Jürgen Wendel. Berlin: Springer. Schouten, Jan Arnoldus. 1922. Über die verschiedenen Arten der Übertragung in einer ndimensionalen Mannigfaltigkeit, die einer Differentialgeometrie zugrundegelegt werden kann. Mathematische Zeitschrift 13:56–81. Seelig, Carl. 1960. Albert Einstein: Leben und Werk eines Genies unserer Zeit. Vienna: Europa Verlag. Speziali, Pierre, ed. 1972. Albert Einstein-Michele Besso: Correspondance 1903–1955. Paris: Hermann. Tonnelat, Marie Antoinette. 1966. Einstein’s Unified Field Theory. New York: Gordon & Breach. Verhaegh, Sander. 2020. Coming to America: Carnap, Reichenbach, and the Great Intellectual Migration. II: Hans Reichenbach. Journal for the History of Analytical Philosophy 8:11. Vizgin, Vladimir Pavlovich. 1994. Unified Field Theories in the first Third of the 20th Century. Translated by Julian B. Barbour. Boston, Basel/Stuttgart: Birkhäuser. Weyl, Hermann. 1918a. Gravitation und Elektrizität. Sitzungsberichte der Preußischen Akademie der Wissenschaften, 465–480. Repr. in Weyl, 1968, Vol. 2, Doc. 31. ——. 1918b. Raum–Zeit–Materie: Vorlesungen über allgemeine Relativitätstheorie. Berlin: Springer. ——. 1918c. Reine Infinitesimalgeometrie. Mathematische Zeitschrift 2:384-411. Repr. in Weyl, 1968, Vol. 2, Doc. 30. ——. 1919a. Eine neue Erweiterung der Relativitätstheorie. Annalen der Physik, 4th Series, 59:101–133. Repr. in Weyl, 1968, Vol. 2, Doc. 34. ——. 1919b. Raum–Zeit–Materie: Vorlesungen über allgemeine Relativitätstheorie, 3rd ed. Berlin: Springer. Weyl, Hermann. 1920a. Die Diskussion über die Relativitätstheorie. Die Umschau 24:609–611. ——. 1920b. Elektrizität und Gravitation. Physikalische Zeitschrift 21:649–650. Repr. in Weyl, 1968, Vol. 2, Doc. 40. ——. 1921a. Das Raumproblem. Jahresbericht der Deutschen Mathematikervereinigung 30:92– 93. ——. 1921b. Electricity and Gravitation. Nature 106:800–802. Repr. in Weyl, 1968, Vol. 2, Doc. 48. ——. 1921c. Feld und Materie. Annalen der Physik, 4th Series 65:541–563. Repr. in Weyl, 1968, Vol. 2, Doc. 47.

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——. 1921d. Raum–Zeit–Materie: Vorlesungen über allgemeine Relativitätstheorie, 4th ed. Berlin: Springer. ——. 1921e. Über die physikalischen Grundlagen der erweiterten Relativitätstheorie. Physikalische Zeitschrift 22:473–480. Repr. in Weyl, 1968, Vol. 2, Doc. 46. ——. 1921f. Zur Infinitesimalgeometrie: Einordnung der projektiven und konformen Auffassung. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 99–112. Repr. in Weyl, 1968, Vol. 2, Doc. 43. ——. 1923. Mathematische Analyse des Raumproblems. Vorlesungen gehalten in Barcelona und Madrid. Berlin: Springer. ——. 1924. Review of Reichenbach, Axiomatik der relativistischen Raum-Zeit-Lehre [Reichenbach, 1924]. Deutsche Literaturzeitung 45:2122–2128. ——. 1929. Elektron und Gravitation. Zeitschrift für Physik 56:330–352. Repr. in Weyl, 1968, Vol. 3, Doc. 85. ——. 1931. Geometrie und Physik. Die Naturwissenschaften 19:49–58. Repr. in Weyl, 1968, Vol. 3, Doc. 93. ——. 1968. Gesammelte Abhandlungen, ed. Komaravolu Chandrasekharan, Vol. 4. Berlin: Springer. Wünsch, Daniela. 2005. Einstein, Kaluza, and the Fifth Dimension. In Kox and Eisenstaedt, 2005, 277–302.

Chapter 7

Special Relativity from the Viewpoint of R. W. Sellars’ The Philosophy of Physical Realism Matthias Neuber

Abstract Roy Wood Sellars (1880–1973) is often reduced to his role as father of Wilfrid Sellars. This is unfair because during the 1920s, ‘30s, and ‘40s, Roy Wood was one of the leading figures of the then prevailing American realist movement. In the present paper, I will focus on one particular facet of R. W. Sellars’ philosophical approach: his continual examination of Albert Einstein’s special theory of relativity. I shall primarily reconstruct his discussion of Einstein’s theory, as it can be found in his seminal The Philosophy of Physical Realism (1932). In contrast to authors such as Bertrand Russell or Émile Meyerson, Sellars refused to interpret special relativity in a realist vein. In his view, it should be seen as an “ars mensurandi” and thus being interpreted purely operationally. As with Einstein himself, the concept of simultaneity was his paradigm case in point. However, Sellars opined that besides the physical (mensurational) concepts of time and simultaneity there also exists an ontological understanding of these notions. “Real” time and “absolute” simultaneity are, according to Sellars, the indispensable nonrelativistic counterparts to Einstein’s respective relativistic conceptions. They are to be interpreted realistically since they prove, Sellars maintains, to be explanatory regarding events in Einstein-Minkowski’s world. In the course of the paper, I shall compare this view with the one defended by Henri Bergson. Furthermore, Sellars’ later approach from the 1940s and ‘50s will be briefly considered and critically discussed by confronting it with more recent attempts at ontologically ‘grounding’ special relativistic kinematics. Keywords Roy Wood Sellars · Physical realism · Special relativity · Simultaneity · Henri Bergson · Real time

M. Neuber () University of Mainz, Mainz, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Russo Krauss, L. Laino (eds.), Philosophers and Einstein’s Relativity, Boston Studies in the Philosophy and History of Science 342, https://doi.org/10.1007/978-3-031-36498-3_7

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7.1 Introduction Roy Wood Sellars received his Ph.D. in philosophy in 1908 from the University of Michigan. Actually Canadian-born, Sellars became one of the most outstanding exponents of American ‘critical’ realism around 1920. In the 1930s, ‘40s, and ‘50s he repeatedly published papers and chapters on Albert Einstein’s special theory of relativity. However, as early as 1908, Sellars had already come out with a twopart article on “Critical Realism and the Time Problem.” Although not mentioning Einstein there, some of the key ideas of his later interpretation of Einstein’s theory can be found in this early article. Therefore, I will begin my analysis with a short glimpse on it (Sect. 7.2). Then, the most important general features of Sellars’ seminal The Philosophy of Physical Realism from 1932 will be briefly outlined (Sect. 7.3). In a next step, I shall say a few words about Sellars’ purely operationalist interpretation of special relativity, as it can be found in The Philosophy of Physical Realism (Sect. 7.4). After that, his presentation of a physical realist ontology of “real” time and “absolute” simultaneity will be discussed at some length (Sect. 7.5). A longer digression on Sellars’ relation to the contemporary view of Henri Bergson will follow (Sect. 7.6). Finally, Sellars’ later approach from the 1940s and ‘50s will be briefly considered and critically discussed by confronting it with more recent attempts at ontologically ‘grounding’ special relativistic kinematics (Sect. 7.7). A short summary will conclude the paper (Sect. 7.8).

7.2 Sellars’ Earliest Writings on Critical Realism and the Time Problem Sellars’ two-part article “Critical Realism and the Time Problem” appeared in 1908 in The Journal of Philosophy, Psychology and Scientific Methods (nowadays The Journal of Philosophy). Founded in 1904 under the editorship of Frederick J. E. Woodbridge and James McKeen Cattell, this journal proved to be a stronghold of the emerging American realist movement in the early twentieth century. According to James Campbell (Campbell 2007, 3) realism was “the primary perspective in American philosophy after about 1900” (2007, 3; see further Campbell 2006, ch. 6). At any rate, Sellars was one of the main promoters of (what, in fact, he initially called) ‘critical’ realism.1 Thus at the very beginning of the first part of his 1908 article, he makes it clear that his aim is twofold: first, he wants to show, as it were, directly that contemporary idealism (as represented, for instance, by F. H. Bradley, A. E. Taylor and Josiah Royce) is unable to cope with the “metaphysical difficulties” (1908a, 542) associated with the time problem; second, he seeks to “indicate the

1 For

a comprehensive display of early twentieth century (‘new’ and ‘critical’) American realism, see Werkmeister 1949, chs. 17 & 18. See further Montague 1937 and Schneider 1964.

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answer of critical realism” (ibid.). Unfortunately, Sellars does not further specify what he understands by ‘critical realism.’ However, for the time being, it is worth noting some of the key points of his anti-idealist approach toward the time problem. To begin with, as Sellars sees it, a principled distinction must be drawn between “real” (or “stereometrical”) time, on the one hand, and “constructed” (or “linear”) time, on the other. While the latter implies a distinction between past, present and future, the former implies a block-like whole of “dynamic relations” (1908a, 547). While constructed time pertains to “experience” (ibid.), real time (unsurprisingly) pertains to experience-independent “reality” (ibid.). And here is where Sellars’ critique of idealism comes into play. He writes: “Space forbids a critical study of absolute idealism, but this I will say: that disregard of, or lack of explanation for, the patent facts of our experience is characteristic of it, and any position that acknowledges and explains these facts is, therefore, preferable.” (544) No wonder, then, that critical realism with its postulation of “real” time is praised by Sellars as the sought-for explanatory position regarding the time problem. Now, in the second part of his 1908 article, Sellars provides a sketch of how “the individual’s time-experience and time-construction” (1908b, 597) can be explained on the basis of the (metaphysical) notion of “real” time and the respective “process view” (ibid.). As he makes it plain at the very end of the article, constructed time “has no metaphysical, but only a logical, significance for critical realism” (602). Again, the individual’s experience is categorically contrasted with “reality as a larger process” (ibid.), thereby implying that this larger process includes the individual and its particular time experience and time construction. Crucial here is the assumption of the individual’s mind as being embodied. Thus, some pages earlier, Sellars declares: “the relation of the individual’s experiencing to the rest of him which we call his body, as a part of reality, is the vital metaphysical problem and the key to critical realism” (598). In other words, the individual, conceived as embodied mind, forms part of the encompassing world of bodies and is thus subjected to the reign of “real” time. So much for Sellars’ earliest account of time and realism. Yet, as already stated, Sellars does not provide us with a definition (or, at least, a reference to one) of critical realism in his 1908 publication. Astonishingly enough, pretty much the same holds true regarding his 1916 programmatic monograph Critical Realism: A Study of the Nature and Conditions of Knowledge. After all, as the subtitle indicates, critical realism is primarily concerned with knowledge and is thus an epistemological position. On the other hand, Sellars is eager to stress the abductive, experiencetransgressing, aspect particularly of scientific knowledge. Thus he points out: To be understood properly, Critical Realism must be connected with a non-apprehensional view of knowledge. Scientific knowledge about the physical world consists of propositions which do not attempt to picture it. It is upon this principle that I take my stand. These propositions must be tested immanently or within experience, but, after being so tested, they are considered as being knowledge about that which can never be literally present within the field of experience, although it controls the elements in the field. (Sellars 1916, vi–vii)

In his 1922 Evolutionary Naturalism, Sellars becomes more concrete as to ‘essence’ of critical realism. In point of fact, he gives the following neat charac-

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terization of it: “It is realism because it maintains that the human mind can build up knowledge about extramental realities; and it is critical realism because it holds that these realities cannot be presented to an immediate awareness, as naïve realism inclines to assert.” (Sellars 1922, 20) Accordingly, the critical realist stance amounts to the view that knowledge in general is both direct and mediated: it deals not with percepts or sense-data, but with mind-independent existing things; though this only via concepts, ideas, and other ‘interpretative tools.’ Around 1920, critical realism, understood thusly, was the prevailing movement in the American philosophical arena. Among the other advocates of this position were thinkers such as George Santayana, Durant Drake, Arthur O. Lovejoy, and C. A. Strong.2

7.3 Physical Realism as the Ontology of Critical Realism We now come to the principal reference point of our examination, Sellars’ seminal The Philosophy of Physical Realism (henceforth PPR). Published in 1932, this volume marks, as it were, the next step in Sellars’ philosophical development. Werkmeister writes in this connection: As the controversy over epistemological realism draws to an end, some of the realists, regarding the problems of epistemology as settled in their favor, turned to discussions of metaphysical issues and to the development of comprehensive philosophical systems. Foremost among these thinkers is Roy Wood Sellars [ . . . ]. (Werkmeister 1949, 486)

And indeed, already in the preface to PPR, Sellars leaves no doubt about his major objective. He frankly states: “I am an unashamed ontologist and a convinced believer in the ontological reach of science. And this in spite of pragmatist, Viennese positivist, or religious personalist. If empirical knowledge is not knowledge of what exists, then it is not knowledge.” (Sellars 1932, vii) Thus, in methodological terms, Sellars (in contrast to, say, Rudolf Carnap) refused to reject metaphysics, and in particular ontology, as ‘meaningless.’ More specifically, he refused to take the linguistic turn.3 Now in chapter III of PPR, titled “The Return to Realism,” Sellars recaps his 1922 characterization of the central thesis of critical realism as follows: “Knowing is direct in that its primary object is objective disclosure; but it is mediated by data and concepts.” (1932, 61) Moreover, in chapter IV, Sellars goes one step further,

2 For

further details of the impact of the critical realist movement, see Hatfield 2015 and Neuber 2020. For its presumably German roots, see Neuber 2017. 3 Thus, in his retrospective Reflections on American Philosophy from Within, Sellars reports: “[S]ocalled analytic philosophy [ . . . ] did not seem to me very creative in either epistemology or ontology. American addiction to it and disregard of its own momentum struck me as a form of neo-colonialism.” (Sellars 1969, 5)

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claiming that “[e]pistemology does not work in a vacuum. To understand human knowing we must use what is known.” (63) Accordingly, the position of physical realism is destined to spell out the ontological presuppositions of critical realism. It provides, in short, the ontology of critical realism. To be more precise, physical realism entails a commitment to the existence of theoretical entities. In Sellars’s own words: “For us, atoms, electrons and photons are as real as chairs and tables.” (97) As he further elucidates, scientific knowledge finds its peak in the performance of measurements. Sellars writes: The scientist makes observations in order to get data. He extends the range and depth of sense-perception by means of instruments. He adopts the method of measurement and gets things to speak to him in terms of each other. He builds up explanatory hypotheses with respect to what is out there in the physical world. (98)

Let us keep in mind: according to physical realism, scientific knowledge transcends mere sense perception, reaches out for theoretical entities, and attempts to ‘tame’ these hypothetically postulated entities by instrument-based testing and measurement. So, how is special relativity to be interpreted within a physical realist framework? One might suspect that Sellars goes for a straightforward ontological interpretation of this theory. But, as we will see in a moment, this is definitely not the case.

7.4 Special Relativity as ars mensurandi Again, in the preface to PPR, Sellars commits himself to an “operational interpretation” (viii) of the theory of relativity. For him, this theory is nothing but “a theory of measurement” (ibid.). Sellars explains: It does not have the ontological significance usually given to it by those philosophers who have become worshipers in the courts of mathematics and physics. [ . . . ] I presume that the positivistic scientist who is afraid of ontology and always adds that he speaks as a physicist – whatever that may mean – may take the theory of relativity as more than a theory of measurement; but I do not see how a physical realist can. (ibid.)

By attentively reading Sellars’ book, one will realize that it is primarily Bertrand Russell’s interpretation of Einstein’s theory (see Russell 1926) from which Sellars is demarcating himself here. We will come back to this point later. For the moment, however, it is interesting to note that, in contrast to the theory of relativity – which according to Sellars “must be given only epistemic significance” (1932, 24) –, quantum mechanics indeed has for him ontological significance. He writes: “[T]he quantum theory is more ontological and, in the strict sense, explanatory than the theory of relativity. It is to be noted that it works with protons, electrons, atoms, and atoms of action. It is, I believe explanatory and ontological like these.” (298) Thus modern physics – which Sellars maintains “[w]e might almost define [ . . . ] as measurement” (241) – is divided by a clash of ontological commitments: while quantum mechanics stands in need of a realistic interpretation of its measurement

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outcomes,4 the theory of relativity is a pure theory of measurement or, better, measurement operations. Hence Sellars’ comment that “I have only admiration for relativity as a theory of measurement but reject it as an ontology.” (356). Let us now come to Sellars’ interpretation of special relativity, in particular. The relevant textual basis here is chapters XI (“Physics and Ontology”), XII (“A Defense of Substance”), and especially XIII (“The Ontology of Space and Time”) of PPR. As Sellars aptly remarks in chapter XI, special relativity for him is an “ars mensurandi” (259). It is thus a theory of mensurational operations and nothing else. Sellars’ paradigm case in point is, as with Einstein himself, the notion of simultaneity of distant events. As is well known, relative movement together with the constant velocity of light involves, according to Einstein, differences in measurement. Consequently, simultaneity is not absolute anymore (like in Newton’s Principia) but relativized to observers and their respective measurement procedures. It is for this reason, that simultaneity is defined operationally. Or, as Einstein puts it in his 1905 “On the Electrodynamics of Moving Bodies:” We have not defined a common ‘time’ for [the spatially separated points] A and B, for the latter cannot be defined at all unless we establish by definition that the ‘time’ required by light to travel from A to B equals the ‘time’ it requires to travel from B to A. (Einstein [1905] 1923, 40)5

The corresponding definition of simultaneity entails then the need of synchronizing clocks at different places by means of light-signal exchanges. Or, as Sellars comments on this issue: In order to correlate [ . . . ] clocks, time-signals must be used; and the logical signal is the light-signal. But it is soon discovered that, by means of light-signals, only relative, and not absolute, simultaneity can be attained. This means that the clocks will not agree when they are on inertial systems moving relatively to one another. Here is the root of the theory of relativity. (1932, 264)

I dare say this is an entirely adequate characterization of the special relativistic scenario. However, Sellars goes one step further, offering the following – nontrivial – philosophical interpretation: The first step taken by Einstein was to define simultaneity operationally. The rest followed. But philosophy may well refuse to accept for its purposes such an operational definition. It is concerned with real time and with the categorial constitution of nature and not with measurement. And yet we must also appreciate what the physicist is doing and what kind of knowledge he obtains. (264)

“Real time,” as already postulated in the 1908 article, is reappearing here. Now we find it contrasted with the Einsteinian conception of time, simultaneity, and the interdependence of time and space in concrete measurements. Notice that

4 This, of course, implies a commitment to some form of de Broglie-Bohm mechanics (as contrasted

to the prevailing Copenhagen interpretation). quote here from the slightly corrected translation in Grünbaum 1973, 343–4.

5I

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Sellars does not intend to supplant the Einsteinian conception. Rather, his aim is to establish a two-aspects view: as he sees it, the interdependence of space and time in measurement – the operational aspect – does not mean that space and time as categories – the ontological aspect – have fused. What is more, in Sellars’ view, real time serves as the ontological foundation of measured (“chronological”) time. Accordingly, operationalism at the physical-descriptive level is supplemented by realism at the philosophical-explanatory level. No doubt this sort of ‘ontological foundationalism’ needs to be carefully analyzed. In point of fact, it will concern us in the remainder of this paper.

7.5 “Real” Time and “Absolute” Simultaneity It is important to note that by “real” time Sellars does not mean Newtonian time. Already in the 1908 paper one can find the claim that “real time is identifiable with change” (1908a, 547). This is surely at odds with Newton’s conception of absolute time as a static ‘container’ inside which changes and events take place. In PPR, then, Sellars argues that real time is a feature of reality which expresses itself in motion and thus cannot exist apart from events. He points out: Time is but a term for changes, and changes are always local and immersed in what is changing. The philosopher has got rid of the tyranny of time as something external to the world. Time is in the world and not the world in time. Events constitute real time. Were there no events there would be no time. And space scatters events. It makes them a manifold and plural in locus as it is. (1932, 261)

It is in chapter XIII of PPR that Sellars spells out the details of his “real time”-conception. His central assumption there is that the physicists’ mensurational operations essentially presuppose a spatio-temporal constitution characterized by “intrinsic determinateness” (311). Otherwise it would make no sense to measure at all. Sellars explicitly rejects any “receptacular” (ibid.) and thus Newtonian notion of real time. Rather, real time for him is “adjectival” (ibid.), having no existence apart from the physical universe. It is this adjectival, event-dependent form of real time which, according to Sellars, lies at the bottom of all measurement. In his own words: Now, is there any reason to relinquish these natural assumptions [regarding real time] because measuring from different frames moving with respect to one another are relative? I cannot see it. Hence, I conclude that the ontological import of the theory of relativity has been quite exaggerated. It is not Space and Time as categories which have been altered but space and time as the quantities s and t which have been shown to be interdependent. It is my thesis that even such relative measurement presupposes the determinateness of what is being measured. (312)

It comes as no surprise that real – adjectival – time, which in contrast to Newton’s absolute time is not global but only local, serves (together with ‘eventscattering’ space) as the ontological guarantee of determinateness in measurement. “Our measurements,” Sellars writes, “may well be relative but it does not follow that

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what is measured is relative. To hold otherwise appears to me a case of operational abstraction very akin to idealism.” (311) So what about simultaneity? It is as early as in chapter XII (which deals with substance) that Sellars categorically claims: Absolute simultaneity must be thought in terms of classes and not of relations. There is no relation of simultaneity and no relation, but only order, of before and after. This substitution of order for relation is the clue to the ontology of time. Strangely, Russell has not seen it. Hence he takes relativity, not merely as an affair of physics, but also as an affair of ontology. [ . . . ] Again, natural categories go by the board without hesitation. This is the mistake of over-sophistication. (277)

Accordingly, absolute simultaneity – as contrasted to Einsteinian simultaneity – must be conceived in non-relational terms, that is, as the block-like class of coactual events. As with the 1908 article, past, present and future are seen as mere ephemeral (relational) constructions. Thus later – in chapter XIII – Sellars specifies: Absolute simultaneity is an affair of order and not of relations, for I do not believe in temporal relations. Since the past has perished, for instance, how can it be literally related to the present? And a similar objection holds for the future. Simultaneous events are not related but are in the class of co-actuals as against events which have perished and those which have not yet arisen. (314)

Consequently, absolute simultaneity, as the principal implication of real time, forms the essential building block of the ontological – physical realist – ‘grounding’ of special relativistic kinematics. Let me add a few words about Sellars’ critique of Russell. What does he mean when he says, in the quotation above, that Russell takes relativity “also as an affair of ontology?” In order to answer this question, one must take a look in Russell’s 1926 entry on relativity for the Encyclopedia Britannica. There one can find a section titled “Realism in relativity.” For Russell, a realistic, i.e. ontologically affirmative, interpretation of relativity is the only way to go. He writes: It is a mistake to suppose that relativity adopts an idealistic picture of the world – using “idealism” in the technical sense, in which it implies that there can be nothing which is not experience. The “observer” who is often mentioned in expositions of relativity need not be a mind, but may be a photographic plate or any kind of recording instrument. The fundamental assumption of relativity is realistic, namely, that those respects in which all observers agree when they record a given phenomenon may be regarded as objective, and not as contributed by the observers. (Russell 1926, 332)

Interpreted that way, relativity theory is less relative than commonly supposed. And indeed, Russell’s interpretation comes pretty close to Einstein’s own account of the philosophical implications of his theory. Thus in his 1928 review of Émile Meyerson’s La Déduction relativiste (1925), Einstein approvingly comments on Meyerson’s realistic approach as follows: Pure positivism and pragmatism are rejected, indeed ardently combatted. Subjective experiences or facts of experience are indeed the basis of every science, but they do not make up its content, its essence; rather they are merely the given to which science refers. Merely ascertaining empirical relationships between experimental facts cannot, according to the author, be represented as the only goal of science. First of all, such general relationships as

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are expressed in our laws of nature are not at all the mere ascertaining of the experiential; they can be formulated and derived only on the basis of a conceptual construction which cannot be extracted for us from experience as such. Secondly, science by no means contents itself with formulating laws of experience. It seeks, on the contrary, to build up a logical system, based on as few premises as possible, which contains all laws of nature as logical consequences. This system, or rather the structures occurring in this system, is coordinated with the objects of experience; reason seeks to arrange this system, which is supposed to correspond to the world of real things of prescientific Weltanschauung, in such a way that it corresponds to the totality of facts of experience or (subjective) experiences. Thus, at the base of all natural science lies a philosophical realism. (Einstein [1928] 1985, 252–3)

Marco Giovanelli has recently shown that Einstein thought of Meyerson’s approach as highly instructive (see Giovanelli 2018). Just as with Russell, Meyerson conceived of the theory of relativity as such as contributing to a realistic worldview. And Einstein obviously agreed. Yet Sellars, on the other hand, speaks of “the mistake of over-sophistication” in this connection. Consequently, he insists on a purely operationalist interpretation of (special) relativity and allocates the job of ontologically ‘grounding’ it to an (allegedly) autonomous instance, namely his own brand of physical realism with its thoroughly anti-relativistic conceptions of real time and absolute simultaneity. Sellars’ foundationalist attitude becomes particularly overt by his comments on Einsteinian inertial frames. On his view, what is needed is a non-relativistic categorial framework which first of all enables measurements. Inertial frames do not suffice in this regard. Sellars explains: “The Einsteinian conception of frames of reference presupposes a universe in which they are established, for these frames are not imaginary but real. In order to move in relation to one another they must be somehow coexistent and in the same universe.” (1932, 311) In order to adequately understand Sellars’ major point, allow me to quote the following longer passage: Returning to the special theory, we would point out again that the art of physical measurement was confronted with the fact that systems moving with respect to one another could not have the same metrics, since their simultaneities, operationally defined, would not be the same. Hence the quantities obtained and entering in judgment about events would each be significant only for a particular frame. Fortunately, an equation could give the proper transformation for the other frame. It meant that if c, the velocity of light, were taken as constant and the same for every frame – and there was no reason for preference since that would introduce an element of absoluteness – clocks would be judged to go more slowly for a frame moving with respect to our own. Our own interpretation of this is epistemic. Einstein’s disciples go beyond this modest outlook, however, and want to transform Space and Time and not merely s and t. They want the clocks to go actually slower, to be retarded locally and in existence and not merely to be read off from another frame as going slower. In other words, they want it to be ontological and not merely epistemic and metric. It is to this interpretation that I object. (312–3)

A central consequence of the (direct) ontological interpretation of special relativity is that there isn’t any “absolute referent for measurement upon which physicists can agree” (313) anymore and that all such agreement becomes “merely a matter of convention” (ibid). But what about the Lorentz transformations to which Sellars is apparently alluding in the quote above? Are they not designed to account for the transformation from one inertial system to another? And are

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they not the guarantee for what Russell deals with under the rubric of objectivity? Sellars does not deny this. But he wants more. As he laconically comments: “I am quite old-fashioned. As a realist, I seek more than quantities and equations.” (320) The Lorentz transformations are for him merely a helpful mathematical device for attaining certain epistemic ends. To have ‘ontologized’ these epistemic ends is, according to Sellars, the capital mistake of Russell and other of “Einstein’s disciples.” The Lorentz connection is quite elucidating. Toward the end of chapter XIII, Sellars summarizes his previous discussion as follows: “When all is said and done, I cannot see that the theory of relativity implies any profound change in our usual categories. Its significance is epistemic rather than ontological. And so I sympathize with the hesitations of Lorentz in regard to all but ether.” (330) As is well known, Lorentzian relativity has a wider explanatory range than its Einsteinian counterpart since it employs classical conceptions such as absolute space and time, absolute motion, and particularly the ether. Regarding simultaneity, Lorentz himself at one place declares: According to Einstein it has no meaning to speak of motion relative to the aether. He likewise denies the existence of absolute simultaneity. It is certainly remarkable that these relativity concepts, also those concerning time, have found a rapid acceptance. The acceptance of these concepts belongs mainly to epistemology. [ . . . ] It is certain, however, that it depends to a large extent on the way one is accustomed to think whether one is attracted to one or another interpretation. As far as this lecturer is concerned, he finds a certain satisfaction in the older interpretations, according to which the aether possesses at least some substantiality, space and time can be sharply separated, and simultaneity without further specification can be spoken of. In regard to this last point, one may perhaps appeal to our ability of imagining arbitrarily large velocities. In that way, one comes very close to the concept of absolute simultaneity. (Lorentz, quoted from Craig 2008, 15)

The assumption of an ether notwithstanding, Lorentz argues here in the same way as Sellars does in PPR: the Einsteinian relativistic conceptions are relegated to the realm of epistemology, and absolute simultaneity is uphold. Yet, one might wonder on which methodological basis both Sellars and Lorentz think of themselves as being justified in postulating an ontology of real time and absolute simultaneity. Is it more than mere speculative metaphysics they are striving for? Before we turn to addressing this question it is worth weaving in a digression on contextual information concerning Sellars’ relation to the contemporary interpretation provided by Henri Bergson.

7.6 Sellars’ Relation to Bergson’s Point of View The academic year of 1909–10 Sellars spent in Europe, particularly in France and Germany. Interestingly, it was in Paris that he studied with Henri Bergson (see Warren 1975, 23). Given his 1908 article, it is plausible to assume that Sellars and Bergson also discussed the time problem. But be that as it may, it appears that Bergson, in his 1922 Durée et simultanéité, goes in a similar direction like Sellars.

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In order to better understand this assessment, it is instructive to take a short glimpse on Bergson’s general philosophical outlook (for a reliable comprehensive survey, see Ansell-Pearson 2011). To begin with, Bergson is no friend of scientism. He believes, that is, in the autonomous explanatory power of certain philosophical conceptions. Underlying this belief is a principled distinction between understanding and science, on the one hand, and intuition and life, on the other. What is more, intuition serves according to Bergson as the principal method of metaphysics which, to a large extent, equals philosophy of life in his overall conception. As he sees it, metaphysics pertains to the absolute. By ‘the absolute’ he primarily means time because time, in his view, is the ‘essence’ of life (and of all things). Given these considerations, it is interesting to see that Bergson, in Durée et simultanéité – translated in English as Duration and Simultaneity –, draws a distinction between two irreducible forms of time: duration or real time (“temps reel”) and physical or scientific time (“temps scientifique”). Notice that Einstein’s theory of relativity is lurking in the background here. Hence the book’s subtitle “With Reference to Einstein’s Theory.” Time as duration is characterized by Bergson as “the continuity of inner life,” “succession without separation,” and “flow” (see Bergson [1922] 1965, 44). His central claim is that time as duration (or real time) is fundamental or, as he himself puts it, “basic” (ibid.). Thus he declares: “Such is immediately perceived duration, without which we would have no idea of time.” (ibid.) Moreover, Bergson assumes that real time has no instants. Rather, the idea of instants gets, in his view, formed by “converting time into space” (52). More precisely, it is mathematical symbols that effect “spatialized time” (53) in the context of concrete measurements. As we have seen, much the same is taught by Sellars. Regarding simultaneity, Bergson distinguishes between simultaneity of flow or real simultaneity (“simultanéité réelle”) and simultaneity of the instant or fictive simultaneity (“simultanéité fictive”). Regarding the latter, he points out: “It is [ . . . ] the simultaneity between two instants of two motions outside of us that enables us to measure time; but it is the simultaneity of these moments pricked by them along our duration that makes the measurement of time.” (54). And he immediately adds: “[I]t is clear that the theory of relativity itself cannot help acknowledging the two simultaneities that we have just described; it confines itself to adding a third, one that depends upon a synchronizing of clocks.” (54–5) Moreover, again just as with Sellars, Bergson argues for a foundationalist conception regarding absolute simultaneity. In his own words: “[I]f we did not begin by admitting a simultaneity [ . . . ] which is absolute and has nothing to do with the synchronizing of clocks, the clock would serve no purpose.” (55) Therefore: “We believe that a philosophy in which duration is considered real and even active can quite readily admit Minkowski’s and Einstein’s space-time [ . . . ]. On the other hand, you will never derive the idea of a temporal flow from Minkowski’s schema.” (63) In short, special relativity needs to be grounded by an absolutist account of time and simultaneity. With respect to frames of reference (inertial frames), Bergson opines that the philosopher, in contrast to the physicist, is not obliged to allow for them. In his

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Fig. 7.1 Simultaneity according to Einstein

view, the philosopher can postulate the one “conscious observer” (95). This would be an observer who is present in both of two systems moving relatively to each other in a uniformly unaccelerated manner – in Einstein’s example: a train and an embankment. Bergson suggests to add arrows in the figure below in the opposite direction, that is, from B to A, thereby implying that the train and the embankment are in a state of reciprocal motion (and refraining from Einstein’s choice of the embankment as his frame of reference). M’ is the observer in the train, M the observer on the embankment. Both of them will witness lightning bolts at A and B, M’ in the moving train, M from the embankment (Fig. 7.1). According to Einstein, it would have to be assumed that we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of coordinates, are simultaneous, can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system. (Einstein [1905] 1923, 42–3)

Bergson disagrees. According to him, the imagined one “conscious observer” could account for the “real” situation, that is, for the two absolutely simultaneous events A and B. Bergson writes: “[I]f we really cling to the perceived, to the lived, if we question a real observer in the train and on the track, we shall find that we are dealing with one and the same time – what is simultaneity with respect to the track is simultaneity with respect to the train.” (95) The adding of arrows from B to A is motivated thus: “[I]n marking the double set of arrows, we have given up adopting a system of reference, we have mentally placed ourselves on the track and in the train at one and the same time; we have refused to turn physicist.” (ibid.) It is not hard to imagine that Einstein was not delighted by Bergson’s absolutistic counter theory. On 6 April 1922 the philosopher and the physicist met at the Société franҫaise de philosophie in Paris to discuss the meaning of relativity (see Canales 2005; further Canales 2018). Bergson at one point of the discussion remarked: “All that I want to establish is simply this: once we admit the Theory of Relativity as a physical theory, all is not finished.” (Bergson, quoted from Canales 2005, 1170). Einstein, however, explicitly disagreed, giving philosophy no role in matters of time. In reply to Bergson, he claimed: “The time of the philosophers does not exist, there remains only a psychological time that differs from the physicist’s.” (Einstein, quoted from Canales 2005, 1175–6).6 Einstein’s reply to Sellars would surely have been the same. 6 For a very similar assessment (also explicitly referring to Bergson), see already Schlick 1915, 158.

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However, somewhat surprisingly, in PPR Sellars sharply demarcates himself from Bergson. He points out: “Bergson’s solution lies in his subjectivism. He gives us an accurate intuition of the psychological experience of movement. He shows that mathematics does not do justice to this experience because it leaves mobility out of its abstract picture.” (352) So far so good (for Bergson) – but Sellars continues: “[ . . . ] he does not realize that real, absolute movement is a physical change which must be thought ontologically. As I see it, a movement is a change in the dynamic relationship of enduring things.” (ibid.) Accordingly, Bergson is not ‘ontological enough’ in Sellars’ view. What he says in the following critique of naïve realism would also apply to Bergson’s account: It is undeniable that naïve realism produces an illusion of a disastrous sort. Supposing that the knowing of external things and events is an apprehension of a mysterious and ultimate sort, we regard these external events as simultaneous with our act of cognition. For this reason, we extend the sense of simultaneity, which colors our immediate consciousness, to external events. We cast our Now, which is our sense of actuality, outward across the universe. [ . . . ] But the critical realist looks upon all knowing as an interpretative reference for which the event known need not be simultaneous with the act of knowing, which is a bit of actuality. Thus simultaneity is a problem which transcends our own changing actuality. We have no intuition of the temporal correspondence of outside events with our own conscious acts. (265)

In short, Bergson’s intuitionist “psychologism” (353) and the corresponding one “conscious observer” miss the ontological dimension of absolute simultaneity.

7.7 Special Relativity in Later Sellars The foregoing digression might have helped to better understand the radical aspect of Sellars’ physical realist interpretation of time and simultaneity. One can speak of an uncompromising ontologism here. Psychologically motivated approaches such as Bergson’s fall short in Sellars’ view. Immanently physical interpretations such as Russell’s, too. So the question must be readdressed whether Sellars runs danger of falling victim to speculative metaphysics and thus of fatally losing contact with actual science. PPR does not help further in this regard. We have already exhausted the relevant material it provides. However, the story does not end at this point. In the 1940s and ‘50s, Sellars delivered a whole bunch of articles dealing with Einstein’s relativity. These articles were: • • • • •

“The Philosophy and Physics of Relativity” (1946a) “A Note on the Theory of Relativity” (1946b) “Materialism and Relativity: A Semantic Analysis” (1946c) “Gestalt and Relativity” (1956a) “Physical Realism and Relativity: Unfinished Business” (1956b)

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In view of this list of publications one might get the impression that Sellars was downright obsessed by thinking about relativity. And indeed: next to the mind-body problem, relativity was the issue on which the later Sellars wrote most extensively. In what follows, I will confine myself to the three articles from 1946. Thus, beginning with “The Philosophy and Physics of Relativity,” the first thing to be noticed is that Sellars starts his considerations with the role of “empty space” (or the vacuum) in special relativity. Einstein’s conception of the velocity of light essentially depends on this conception – and Sellars asks: “What is this empty space?” (1946a, 177) His answer: “the ghost of Newton’s reification of mathematical space” (ibid.). Sellars’ rejects this view, arguing instead for an understanding according to which light should not be seen as moving in empty space but rather as an “extensity-activity” involving constant velocity. He writes: [L]ight is an extensity-activity which invariably gives the same velocity-reading in any material system: It is not something moving in space [ . . . ] but is a distance-travelling activity spreading out from its sources. [ . . . ] This position is more radical than Einstein’s since he seems to have relinquished only the stagnant ether to which Lorentz assigned light. In my opinion, in thinking about light it is well to bear in mind the location of the source in a material system. We shall call this the physical locus of a flash and contrast it with any kinematic position assigned to it (ibid.).

In order to substantiate this point of view, Sellars introduces a distinction between relational movement and purely relative motion. While the latter is symmetrical and kinematic, the first is asymmetrical and dynamic. The distance-covering activity of light is, Sellars maintains, “more of the nature of a movement, a process, than a motion in empty space as a frame” (178). In his view, Einstein “tended to confuse the kinematic with the physical.” (184) The physical, for him, is essentially a world of causal processes. Just as with Bergson, he modifies Einstein’s thought experiment of the train passing the embankment such that an absolute observer witnesses the lightning bolts. Sellars explains: If the observer P’ [M’ in Fig. 7.1] is an absolutist he will argue that the flashes from A and B were not seen compresently because his train was moving toward B and away from A and the light had a longer distance to go in one case and a shorter distance in the other. He will explain the fact that he saw the light from B sooner than P [M in Fig. 7.1] did in the same fashion, namely, that the absolute distance the light had to travel to him was less because of his motion. If P is an absolutist he will agree with the argument and hold that the flashes from A and B were absolutely simultaneous. (182)

In Einstein’s relativistic scenario, on the other hand, P’ would argue that the events were not simultaneous for him, but indeed were simultaneous for P. The corresponding relative motions of P’ and P, that is, the inertial frames of P’ and P would be related by the Lorentz transformations. Yet from the absolutistic point of view, relational movement would do the job. Sellars therefore claims: “We do not need the rules of transformation either of the classical sort or the LorentzEinstein kind.” (188) And he specifies: “A material system is not primarily a frame of reference but is so only secondarily and conventionally for descriptive purposes. What happens existentially is a change in distance-relations.” (193) On the whole, then, “Einstein’s is a halfway position.” (ibid.)

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In “A Note on the Theory of Relativity,” Sellars starts with principally opposing space, time, and motion, on the one hand, and extensity, change, and movement, on the other (see Sellars 1946b, 309). He again discusses the example of the simultaneous flashes, contrasts his own “existential analysis” with Einstein’s “kinematic analysis” (see Sellars 1946b, 314–5) and eventually somewhat ridicules the special theory of relativity as “kinematic Berkeleianism” (316). Finally, in “Materialism and Relativity,” Sellars goes as far as to claim that “[my] analysis gives back the one material universe with both a spatial and a temporal unity, whereas the effect of the relativity theory was to break it up into frames of reference, each inhabited by an observer” (1946c, 39). Exploring the details of the later Sellars’ approach to special relativity must be reserved for further study and research. For the time being, though, the reconstruction provided so far might suffice to discuss the critical question raised above, namely whether Sellars’ conception is that of a speculative metaphysician. In order to do so, it is instructive to reflect on the more recent discursive context. Thus in 2008 there appeared an anthology titled Einstein, Relativity and Absolute Simultaneity, edited by William Lane Craig and Quentin Smith. As the editors state in their introduction, almost all the contributors to the volume “are convinced that the received view that simultaneity is not an absolute relation is not only unwarranted but false.” (Craig and Smith 2008, 9) Of course, this is not the place to survey the entirety of articles contained in the volume (they comprise, among others, contributions by Craig Callender, Tim Maudlin, Richard Swinburne, and Michael Tooley). Rather, I will confine myself to the essay provided by co-editor Craig, since his account comes pretty close to Sellars’. According to Craig, “[o]n a space-time ontology, there is [ . . . ] a unified, independent reality which is merely measured differently by observers using different coordinate systems. But on the Einsteinian interpretation, reality literally falls apart, and there is no one way the world is.” (Craig 2008, 23) Accordingly, “the Einsteinian interpretation is explanatory deficient” (ibid.). This is exactly Sellars’ overall position. So what about simultaneity? Again, Craig unmistakably argues in Sellarsian terms, pointing out that “[t]here is a shared, objective reality which exists independently of observers or reference frames, and we all inhabit the same space-time world; we just reckon different events in that one world to stand in the relation of simultaneity with one another” (ibid.). No doubt this sort of simultaneity is absolute and not relative. Now the interesting issue is: How do Craig and Sellars know? What are their criteria for committing themselves to anti-Einsteinian absolutism? In the case of Craig, theological assumptions seem to play a crucial role. Heavily relying on Lorentz’s commitment to a “World Spirit” being able to “directly verify simultaneity” (Lorentz in a letter to Einstein; quoted in Craig 2008, 21), he explicitly states that “the use of light signals to establish clock synchrony is a convention which finite and ignorant creatures have been obliged to adopt, but the living and active God, who knows all, would not be so dependent” (Craig 2008, 21). Thus according to

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Craig “we have powerful metaphysical grounds” (40) for believing in absolutism.7 Yet, for Sellars, the theological way out is definitely not an option. Being signer (and co-author) of the 1933 “Humanist Manifesto” (see Meyer 1982), he, as it were, was obliged to remain within the natural realm.8 But what, then, could be his criterion for absolutism? Hard to say. His above-quoted commitment to an absolute observer seems to be a candidate. But how does this particular observer distinguish from Bergson’s? Would it be a conscious being, then psychologism would come in from the backdoor. Would it not be a conscious being, what would it be then? A machine? Then we would end up with some sort of science fiction. Something super-natural? Then both realism and humanism would be sacrificed. Another option would be to interpret Einstein’s theory itself in absolutistic terms (see, in this connection, for example DiSalle 2006, 109). However, this would stand in direct opposition both to Sellars’ critique of Russell and to his own ontological foundationalism. All in all, then, I see no other way than to conclude that Sellars’ indeed was the position of a speculative (though non-theological) metaphysician. In itself, this is no argument against his account. But it isn’t, I submit, an argument for his particular interpretation of Einstein’s theory either.9

7.8 Summary In the present paper, the not very well-known Einstein interpretation by Roy Wood Sellars has been reconstructed and eventually critically assessed. The focus has been laid upon his account of Einstein’s special theory of relativity, particularly on his attempt at ontologically grounding the Einsteinian conceptions of time and 7 To

be fair, it should be mentioned that in Craig’s view there are also essentially physical criteria for arguing in terms of absolutism, first and foremost criteria pertaining to the situation in general relativity and in (the de Broglie-Bohm conception of) quantum mechanics. The other contributions to the volume focus primarily on these genuinely physical criteria. However, our issue here is special relativity and not these other branches of physics. And in this context I fail to see any other than the theological assumption in Craig’s account. 8 Thus already in PPR Sellars declares: “Humanism, as I profess it, takes nature as ultimate and does not find any of the traditional God-ideas applicable to any part of it or to the whole.” (1932, 236) 9 For reasons of fairness, it should be noted that, very late in his life, Sellars became skeptical about his approach to Einstein’s theory. Thus, in the year before he died, he explicitly confessed: “Since writing the above at ninety-two, I have been reading Einstein’s own account of the development of his theory and find it illuminating. [ . . . ] I had always rejected Newtonian absolute time [ . . . ]. Happenings or events – the essence of time – seemed to me always to have a physical locus. Here I was on the right track. But I did not see clearly that inertial systems moving uniformly created a new problem. It was this circumstance that Einstein emphasized. He showed that one could not synchronize clocks moving rapidly on two different inertial systems – say, at 10,000 miles a minute – with respect to one another. [ . . . ] I had been caught up in the literature of relativity but had not read Einstein’s own book sufficiently. So I find that I was partly right yet also partly wrong.” (Sellars in Warren 1975, Foreword)

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simultaneity. Sellars’ relation to Henri Bergson’s point of view has been considered, connections with more recent approaches have also been taken into account. On the whole, there remain serious doubts regarding the viability of Sellars’ position. But this is not to say that the reconstruction provided here is without alternatives. Rather, it is hoped that interest in interpreting Sellars’ approach to special relativity will develop further.10

References Ansell-Pearson, K. 2011. Henri Bergson: An Introduction. Routledge. Bergson, H. 1922/1965. Duration and Simultaneity. Trans. L. Jacobson. Bobbs-Merrill. Campbell. 2006. A Thoughtful Profession: The Early Years of the American Philosophical Association. Open Court. ———. 2007. One Hundred Years of Pragmatism. Transactions of the Charles S. Peirce Society 43: 1–15. Canales, J. 2005. Einstein, Bergson, and the Experiment That Failed: Intellectual Cooperation at the League of Nations. MLN 120: 1168–1191. ———. 2018. The Physicist and the Philosopher: Einstein, Bergson, and the Debate That Changed Our Understanding of Time. Princeton University Press. Craig, W.L. 2008. The Metaphysics of Special Relativity: Three Views. In Einstein, Relativity and Absolute Simultaneity, ed. W.L. Craig and Q. Smith, 11–49. Routledge. Craig, W.L., and Q. Smith, eds. 2008. Einstein, Relativity and Absolute Simultaneity. Routledge. DiSalle, R. 2006. Understanding Space-Time: The Philosophical Development of Physics from Newton to Einstein. Cambridge University Press. Einstein, A. 1905/1923. On the Electrodynamics of Moving Bodies. In The Principle of Relativity. A Collection of Original Memoirs on the Special and General Theory of Relativity, ed. H.A. Lorentz et al., trans. W. Perret and G.B. Jeffery, 35–65. Dover Publications. ———. 1928/1985. A propos de “La déduction relativiste” de M. Émile Meyerson. In The Relativistic Deduction. Epistemological Implications of the Theory of Relativity with a Review by Albert Einstein, ed. É. Meyerson, trans. D.A. Sipfle and M.-A. Sipfle, 252–256. Reidel. Giovanelli, M. 2018. Physics Is a Kind of Metaphysics: Émile Meyerson and Einstein’s Late Rationalistic Realism. European Journal for Philosophy of Science 8: 783–829. Grünbaum, A. 1973. Philosophical Problems of Space and Time, 2nd enlarged ed. Reidel. Hatfield, G. 2015. Radical Empiricism, Critical Realism, and American Functionalism: James and Sellars. HOPOS: The Journal of the International Society for the History of Philosophy of Science 5: 129–153. Meyer, D.A. 1982. Secular Transcendence: The American Religious Humanists. American Quarterly 34: 524–542. Meyerson, É. 1925. La déduction relativiste. Payot. Montague, W.P. 1937. The Story of American Realism. Philosophy 12: 140–161. Neuber. 2017. Külpe and American Critical Realism. Discipline Filosofiche XXVII: 205–222. ———. 2020. Two Forms of American Critical Realism – Perception and Reality in Santayana/Strong and Sellars. HOPOS: The Journal of the International Society for the History of Philosophy of Science 10: 76–105.

10 Research

for the present paper was founded by the Deutsche Forschungsgemeinschaft, project number 453466855

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