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The Reign of RelativityPhilosophy in Physics 1915-1925$

Thomas Ryckman

Print publication date: 2005

Print ISBN-13: 9780195177176

Published to Oxford Scholarship Online: April 2005

DOI: 10.1093/0195177177.001.0001

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“Critical or Empiricist Interpretation of the New Physics?”

(p.47) 1921
The Reign of Relativity

Thomas Ryckman

Oxford University Press

Abstract and Keywords

Mortiz Schlick’s article of this title was highly influential in convincing several generations of philosophers that GTR outrightly falsified any variety of Kantian epistemology. In fact, the empiricism Schlick countered to transcendental idealism had not yet appeared in his previous writings but was quickly cobbled together from disparate elements: Henri Poincaré’s geometric conventionalism and selective readings of Einstein’s “Geometry and Experience” and earlier texts of Hermann von Helmholtz. The result of Schlick’s improvisation is that the empiricist interpretation of the spacetime metric rests on conventions regarding the behavior of rigid rods and clocks.

Keywords:   Moritz Schlick, general theory of relativity, empiricism, transcendental idealism, Henri Poincaré, geometric conventionalism, Albert Einstein, Geometry and Experience, Hermann von Helmholtz, spacetime metric

O Kant, wer rettet dich vor den Kantianern?

Hans Reichenbach to Arnold Berliner, 22 April 19211

3.1 Introduction

The appearance early in 1921 of Cassirer's “epistemological considerations on Einstein's theory” prompted the editors of the Kant‐Studien, the official organ of the venerable Kant Gesellschaft, Germany's largest and most notable professional association of philosophers, to ask the philosopher Moritz Schlick once again to consider the viability of a Kantian philosophical understanding of the theory of relativity. Already in 1915, just following the appearance of the general theory in November, Schlick had published an assessment of the philosophical significance of the (special) theory of relativity, arguing that Kant's doctrine of time as an a priori form of intuition had been too closely modeled on Newtonian time to be compatible with the new Einstein kinematics. Hence, any claim that space and time are necessary a priori forms of intuition could pertain, at most, to purely qualitative and subjective properties of space and time, not to the quantitative measurable relations of physics. While “not abolishing the core of the Kantian doctrine”, the (special) theory of relativity showed “the necessity of modifying essential parts of it”.2 Now, with the general theory of relativity, Kantian and neo‐Kantian epistemological (p.48) analyses of relativity theory seemed, prima facie, in further difficulty. The Kantian claim regarding the necessarily Euclidean character of space in the doctrine of pure intuition of the Transcendental Aesthetic appeared straightforwardly refuted. Nonetheless, Cassirer, a leading “neo‐Kantian” (although Cassirer himself rejected the label as suggesting a dogmatic attachment to orthodoxy3), had concluded that the theory of general relativity, as evidenced by Einstein's claim that in the new theory space and time lost the “last vestige of physical objectivity”, exhibited “the most determinate application and implementation of the standpoint of critical idealism within empirical science”.4 What were philosophers, excluded from first‐hand knowledge by the theory's highly abstract mathematics, to think?

Cassirer's book merited the editors' special attention for a number of reasons. Recently called, in 1919, as professor of philosophy to the University of Hamburg, newly created by the Weimar government, Cassirer was one of the leading philosophers in Germany.5 He had first made his name in the philosophical world as a historian of philosophy, tracing the “problem of knowledge” from the early Renaissance up through Kant in two large volumes of the same title, in 1906 and 1907. A third volume in the series, covering the post‐Kantian developments of the 19th century, followed in 1920.6 A ten‐volume edition of Kant's works, edited by Cassirer, appeared in 1912; his intellectual biography of Kant completed the edition as eleventh volume in 1918.7 But it was above all the publication of his first book of systematic philosophy in 1910, Substanzbegriff und Funktionsbegriff, that made Cassirer the recognized leader of the second generation of the neo‐Kantian Marburg School. Founded by Hermann Cohen and Paul Natorp, this branch of neo‐Kantianism was most closely associated with epistemological questions of the mathematical and physical sciences.8 As noted in chapter 2, Cassirer's “epistemological considerations” on relativity theory continued and extended the broad theme of that earlier work in which Cassirer tracked the transformation of the concept of object in mathematics and physical science since the 17th century. In the “Forward” to his relativity book, Cassirer reported that Einstein himself had read his book through in manuscript, and “encouraged (it) through several critical comments”.9 Einstein's criticism, expressed in a letter to Cassirer of 5 June 1920, was rather benign, especially when contrasted with his disparaging comment to Schick, in autumn 1919, regarding “how eagerly the philosophers are already striving to cram the general theory of relativity into the Kantian system”.10 To Cassirer, Einstein wrote that he could understand “your idealistic mode of thought” (Ihre idealistische Denkweise) regarding space and time, and even believed it to be free of contradiction, and he likewise agreed that “conceptual functions [begrifflichen Funktionen] must enter into experience in order for science to be possible”. However, he cautioned, the choice of such functions could by no means be thought to be “compelled by the nature of our intellect”.11 Cassirer, as is evident from chapter 2, could only agree.

Schlick's credentials for this assignment were impressive. Schlick was the first philosopher in Germany to be recognized as a competent authority on the theory of relativity. He had received his Ph.D. in physics in 1904 under Max Planck in Berlin with a thesis on the reflection of light in inhomogeneous media that demonstrated Schlick's facility in carrying out detailed scattering calculations.12 Years later, at a celebration on the occasion of his eightieth birthday in 1938, (p.49) Planck would single out, as most notable among his many students, Schlick, together with the Nobel prize winner Max von Laue.13 Still extraordinarius professor in Kiel in 1921, Schlick would go to Vienna in 1922 to become the fourth occupant of the chair in the Philosophy of the Inductive Sciences at the University of Vienna created for Ernst Mach in 1895, and subsequently occupied by Ludwig Boltzmann and the physical chemist L. Stöhr. Besides the 1915 paper on the philosophical significance of (special) relativity principle, Schlick had also written an exposition of the general theory for the scientific laity, Space and Time in Contemporary Physics [Raum und Zeit in der gegenwärtigen Physik], first appearing in the pages of the scientific weekly Die Naturwissenschaften in two installments in March 1917.14 Reprinted as a separate monograph, the third (1920) edition was, in 1921, still on the booksellers' shelves. (As will be seen, the fourth and final edition appeared in 1922, with a significant change.) Finally, Schlick had known professional and personal connections to Einstein; there was considerable correspondence between them in the period between 1915 and the early 1920s.15 Their first correspondence, upon Einstein's reading of Schlick's initial philosophical appraisal of special relativity in 1915, revealed Einstein to be an enthusiastic supporter of Schlick's philosophical writings on relativity theory. In late 1919, he was Schlick's house guest in Rostock, following which, he wrote to Max Born, “Schlick is a clever person [ein feiner Kopf]; we must try to obtain a professorship for him”, a task that would not be easy, Einstein judged, because Schlick “does not belong to the philosophical church of the Kantians”.16 It is likely that Einstein played some role in obtaining Schlick's post in Vienna. In any case, Schlick's large 1918 book on general epistemology, Allgemeine Erkenntnislehre, the first number of a series of monographs and textbooks in the natural sciences published under the editorial direction of the editors of Die Naturwissenschaften, was well known to Einstein. Preparing to travel to Holland in October 1919, Einstein wrote to Schlick that the only reading for the journey was to be his “epistemology”.17 This reading had a visible influence in Einstein's widely read essay “Geometry and Experience” (“Geometrie und Erfahrung”), originating as a rare public lecture on 27 January 1921. There, Einstein not only commended the book's epistemological emphasis of the method of implicit definition, he also wrote of a “geometrical‐physical theory” as “necessarily unintuitive, a bare system of concepts”, the geometrical and some of the physical laws of which are conventions, whose relation to “experiential objects of reality (experiences)” is one of “coordination”, all views found in Schlick's book.18 Perhaps having this reference in mind, the mathematician Hermann Weyl lamented to Edmund Husserl in March 1921 (see chapter 5, §5.2.1), that Schlick's epistemology book had a considerable resonance with “the leading theoretical physicists”. The editors of the Kant‐Studien could not possibly have regarded Schlick as neutral toward the Kantian or “critical” (“kritizistische”) philosophy, for his epistemology book pointedly defended, in explicit opposition to all varieties of Kantianism (including Husserlian phenomenology) and Machian positivism, a form of scientific realism. Nonetheless, Schlick, as both philosopher and Fachmann regarding the theory of relativity, could be regarded as uniquely placed to assess what would be the most philosophically sophisticated attempt to link the general theory of relativity with the broad trend of Kantian thought.

(p.50) In the event, Schlick's review essay appeared in mid‐1921 in the Kant‐Studien. Its title, echoed in that of this chapter, directly revealed his understanding of his assignment. It was to render an answer to an either/or interrogative, tertium non datur. Posed in this way, there could be little surprise regarding his verdict, although garden‐variety empiricists might not have recognized Schlick's “empiricism” as falling quite within any of the known species of that doctrine. Let us now turn to a recent, and authoritative, assessment of the significance of Schlick's answer:

It is the first clear statement of the inconsistency between Kantian philosophy and relativity. This remarkable article may well be regarded as the point of departure of a new direction for scientific philosophy.19

This appraisal, by the late Alberto Coffa, can be accepted as the “received view” of Schlick's essay. There are two claims here, that the “Kantian philosophy” is inconsistent with the theory of relativity, and that Schlick opened up a new direction for scientific philosophy. Certainly both claims are of interest. Although in this chapter I am principally concerned with the latter claim and, in particular, the nature of the empiricist interpretation of the new physics that Schlick offered in place of Cassirer's kritizistiche interpretation, a few words of reminder about the former claim are in order. First of all, thanks in large measure to Schlick's authority and rhetorical ability to pose the issue on his own terms, the debate between “empiricist” and “critical” philosophy over relativity theory effectively ended with Schlick's essay. Within a few years, the postulate that physical theories neither require nor contain any “constitutive” or synthetic a priori elements would become a cornerstone of the new “scientific philosophy” of logical empiricism, as would also the polemical tarring of the “philosophy of the synthetic a priori” with the broad brush of “metaphysics”. Second, if consideration is limited to those few who in 1921 had demonstrated an innovative expertise with the new theory that Schlick himself certainly did not possess, this outcome, so influential for philosophy of science in the 20th century, was by no means inevitable. Subsequent chapters will consider at length the philosophical standpoints of Weyl and Eddington, both kindred spirits to the “philosophy of the synthetic a priori”. Here I simply note the assessment of Schlick's cohort, Max von Laue, made also in 1921 in the first edition of his well‐regarded text on general relativity:

It is, frankly, an identifying characteristic [Kennzeichen] for a correct epistemology, that it remains invariant against all transformations that the physical world picture experiences in the course of time. We would not conceal our conviction that Kant's critical idealism (although not every sentence of the “Critique of Pure Reason”) satisfies this requirement even against the general theory of relativity.20

Third, and finally, on a careful reading, Schlick's argument bears not upon Cassirer's understanding of the “synthetic a priori” as regulative principles or “rules of the understanding” governing the development of concepts of physical objectivity, but upon a more traditional Kantian conception of apodictically certain and unrevisable principles. Perhaps to avoid tiresome discussions of Kant interpretation, perhaps because he considered it the identifying characteristic of all Kantian (p.51) philosophy, perhaps for rhetorical purposes, Schlick located “the essence of the critical viewpoint” in the claim that the constitutive principles of physical knowledge

are to be synthetic judgments a priori in which to the concept of the a priori inseparably belongs the characteristic of apodeicticity (universal, necessary and inevitable validity).21

The “critical” (i.e., neo‐Kantian) philosopher must maintain this understanding of the synthetic a priori, or else, in Schlick's lexicon, he is no longer a “critical” philosopher. Still more, continuing a reading of Kant given in his Allgemeine Erkenntnislehre (1918), Schlick refused to allow that the Kantian doctrine of “pure intuition” could ever be purged of its psychological trappings and so could not be revised or refined to be a “method of objectivification” in the manner Cassirer had adopted. The gauntlet thus laid down, Schlick had little trouble in dispatching such claims of Cassirer's book as could be represented in this fashion, misleadingly, since Schlick completely ignored the genetic character and historical evolution of the “regulative principles” and “rules of the understanding” that comprised the core of Cassirer's account of the development of the concept of physical objectivity culminating in general covariance. Accordingly, with this declaration the issue is no longer joined, for Cassirer had been denied any possibility of distinguishing his conception of constitutive a priori principles from an orthodoxy that Schlick could easily show was rendered obsolete by the new physics. In one recent assessment, Schlick's “challenge” to Cassirer to produce examples of such unrevisable synthetic a priori principles “represents a fundamental misconstrual of Cassirer's conception of the a priori”.22 In any case, Schlick's traditional reading of the synthetic a priori was not a necessary one, as he himself already knew. The principal thesis of a 1920 monograph from the neo‐Kantian perspective of Schlick's logical empiricist colleague‐to‐be, Hans Reichenbach, denied that apodictic certainty is inseparably attached to synthetic a priori principles. Schlick had reviewed Reichenbach's book and subsequently expended considerable effort, in correspondence with Reichenbach in late November, 1920, arguing that Reichenbach's theory‐relative conception of synthetic a priori principles did not suffice to distinguish them from conventions in the sense of Poincaré.23 Showing some understandable sensitivity on this interpretive point, in his essay Schlick still insisted on this view of the a priori, while stating that his was “an inquiry directed to systematic rather than historical questions”. Thus, he gave himself an easy target indeed.

However, this chapter focuses on Coffa's second claim, although factually it is not in dispute. Rather, my concern here is with the nature of the empiricist interpretation of the new physics pointing a “new direction for scientific philosophy” that Schlick offered in place of Cassirer's “critical idealism”. While polemically counterposed to Cassirer as an interpretation already extant in the literature, in fact Schlick's empiricist interpretation, namely, his empiricist account of the metric of space‐time in the general theory of relativity, was a work in progress, hardly then existing. At the time of Schlick's diatribe against Cassirer it had merely been hinted at in several of the notes and elucidations Schlick appended to two of Helmholtz's papers on physical geometry, republished in 1921 in a new edition of (p.52) Helmholtz's epistemological writings. In turn, these crucial notes and elucidations sought to recast Helmholtz's views on physical geometry in the bright light of Einstein's recent paper on “Geometry and Experience” so as to make Helmholtz appear to be an empiricist precursor of both Einstein and Schlick. By themselves, they provide evidence of a fundamental shift in Schlick's account of the interpretation of the geometry of physical space, differing from that presented in his previous writings on relativity theory. For this reason, 1921 draws our attention as a “pivotal year” for “scientific philosophy” because it brought Schlick's crystallization of the new empiricist interpretation of physics. Based on a highly selective reading of these texts of Einstein, and of Helmholtz, Schlick's new empiricism is an almost “on‐the‐spot” improvisation, seeking to find the resources for an empiricist interpretation of the metric of space‐time in observable facts about measurement bodies and light rays whose fiduciary behavior has been fixed by conventional stipulation.

In what follows, three central aspects involved in the emergence of Schlick's “empiricist interpretation” of the new physics are identified and treated severally. First and foremost is Einstein's well‐known lecture “Geometry and Experience” that, as Schlick is writing against Cassirer, had just recently been published as a separatum from the Proceedings of the Berlin Academy of Sciences. Following Schlick's lead, this article would become virtually a founding hymn of logical empiricism. But Schlick chose to ignore the pro tem character of Einstein's defense of rigid rods and ideal clocks as metrical indicators in the general theory of relativity, a hypothesis Einstein knew to be inconsistent with the spirit, if not the law, of his field equations of gravitation. On the other hand, Einstein's treatment of the line element of the space‐time interval as physically defined by measurements of rigid rods and clocks was his principal weapon against Hermann Weyl's theory of “gravitation and electromagnetism”. As discussed in chapter 4, and in further detail in chapter 6, from Weyl's epistemological vantage point such a stipulation regarding rigid bodies represents the last vestiges of Euclidean “distant geometry” in Riemann's infinitesimal geometry and so is unjustifiable even if it is in accordance with the observed behavior of rigid rods and ideal clocks as metrical indicators in weak gravitational fields. Completely disregarding this argumentative context of Einstein's essay, Schlick interpreted Einstein's provisional endorsement of rods and clocks in the present state of physics as instead a methodological affirmation of the conventionalist underpinnings of a new empiricist realism in physics, resting upon a stipulation regarding rigid bodies.

Schlick employed this selective assessment of Einstein's essay in order to refurbish Helmholtz's epistemological project of attempting to base the geometry of physical space upon “facts” about rigid bodies. Blithely overlooking Helmholtz's own attempt to salvage a modified version of the Kantian theory of space as a form of “outer intuition” in his account of the “facts” underlying geometry, Schlick interpreted Helmholtz as an occasionally naive geometric empiricist whose “greatest epistemological achievement, his theory of space”, once corrected in the light of Einstein's supposed definitional treatment of rigid bodies, is not only plausible for the new physics of general relativity but also “quite certainly true”.24 Yet while celebrating Helmholtz as the Elijah of the new empiricism, Schlick also found it necessary to modify his previous unqualified endorsement of the holist (p.53) conventionalism he had associated with Poincaré, a change that occasioned wider reaching ramifications within his general epistemology. Thus, in order to present the “consistent empiricism” he opposed to the neo‐Kantians, Schlick eliminated the gray area recognized in the first edition of that work between “hypotheses” and “definitions”. In the book's second edition (1925), the classification of types of judgment was revised to feature a sharp distinction between definitions and empirical judgments. This new discrimination was the prototype for the particular version of the analytic/synthetic distinction that became a defining characteristic of logical empiricism and the principal target, in an ironical turn of the wheel of fortune, of Quinean holism.

3.2 A New Empiricism?

What is the central tenet of Schlick's new empiricism? The official view of the Vienna Circle identified it as a renunciation of the conception of synthetic judgments a priori in all “scientific philosophy”.25 However, excoriation of the synthetic a priori can be but a necessary, not a sufficient, characterization, for it does not distinguish the new empiricism from what Schlick called the “extreme empiricism” of Mach, in criticism of which Schlick largely agreed with Cassirer. But although unwilling—as the title of his essay reveals—to allow a third way in the choice between empiricism and the strictly Kantian synthetic a priori, Schlick insisted upon such with respect to the choice Cassirer had posed in 1921 between Machian empiricism (Sensualismus) and critical idealism. There is yet another alternative: it is an empiricism with constitutive principles.

Between the two remains standing the empiricist view, according to which these constitutive principles are either hypotheses or conventions; in the first case they are not a priori (since they lack apodeicticity), and in the second they are not synthetic.

Because of this tertium quid,

a thinker who in general perceives the unavoidability of constitutive principles for scientific experience should not yet on that account be designated a critical philosopher (als Kritizist). An empiricist can, for example, very well recognize the presence of such principles; he will only deny that they are synthetic and a priori in the sense described above.26

As we know, Schlick's novel idea of an “empiricism with constitutive principles” created the broad mold for the logical empiricist or logical positivist analysis of scientific knowledge. To invoke only Coffa's evaluation, “there was no doctrine more central than this to the development of logical positivism in the late 1920s and early 1930s”.27 In a brief time, the language of “constitution” would fade from view, particularly after Carnap's ambitious sketch in 1928 of a “constitution theory” for empirical science based on the type theoretic logic of Principia Mathematica.28 What remained of Schlick's holist conventionalism was the idea that a convention or stipulation, Schlick's surrogate for constitutive principles, must be made in order for a physical theory to acquire empirical content. The requirements (p.54) of empiricism mandated that such conventions, in the form of physical or “coordinative definitions” concern observable objects and processes. In short order, the core thesis of logical empiricism emerged, that all cognitive statements could be factored into purely tautological (or analytic) and empirical (or synthetic) components. This revitalization of Humean empiricism was above all due to Schlick, and from it nearly all the subsequent currents of 20th century philosophy of science might be traced, if only in dialectical opposition.

But consider, for a moment, Schlick's claim that the constitutive principles of scientific theories are located within the alternatives “either hypotheses or conventions”. In order to gauge the significance of this disjunction, some further details of Schlick's general epistemology of science are worth noting, as that position is represented in the first edition of his Allgemeine Erkenntnislehre. For Schlick, the process of cognition is essentially unifying and explanatory; its “great task” is to find out how to use more and more general concepts to designate individual or particular objects. This is all the more the case in the sciences posing the ultimate aim of cognition as bringing the totality of phenomena under a minimum of explanatory principles.29 As I showed in chapter 2, §2.4.1, Schlick's account of the process of cognition is entirely erected upon the concept of “coordination” (Zuordung), the basis of the “merely designative (semiotic) character of thinking and cognition”. In Schlick's exposé, concepts are signs coordinated to objects, taken in the wide sense as including perceived or inferred qualities, properties, and relations. With a gesture to Hans Vaihinger's “Philosophy of the As If”, Schlick maintained that, strictly speaking, concepts are not real, they are “mere fictions”, valuable only for their instrumental role in designation, which is the essence of cognition. Since they are “not real mental forms of any kind”, concepts can be precisely defined through definitions, particular judgments that set up relations among concepts.

What is wanted is a mode of definition compatible with the “purely semiotic” character of scientific cognition and with the character of concepts as mere signs. Here Schlick took Hilbert's axiomatization of Euclidean geometry, in particular, to have indicated “a path that is of the highest significance for epistemology” in emphasizing implicit definition of concepts in mathematics as a method that frees concepts from any nonexplicitly expressed (and so, nonconceptual) trappings of meaning. In mathematics, primitive concepts appearing in the axioms of a theory are implicitly defined by their occurrence in the deductive consequences of the axioms, the sole requirement being the mutual consistency of the axioms that relate concepts to concepts. All verbal concepts may bring with them other, more or less vague, semantic connections and psychological associations. But in mathematics, the method of implicit definition has epistemological significance precisely because whatever intuitive meaning (anschauliche Bedeutung) thereby attaches to such concepts (like “point” or “line”) is “completely unimportant” for the deduction of mathematical theorems. The fundamental innovation of Schlick's general theory of knowledge is then to suggest that the method of implicit definition “is by no means restricted to mathematics but is in principle just as valid for all scientific concepts as for mathematical ones”. In this way, rigorous exactness of thinking is purchased at the cost of “a radical separation” of concepts from intuition, and of thinking from reality (Wirklichkeit).30 On the one side lies a system of scientific (p.55) concepts, precisely defined within a (hopefully) consistent axiomatic system; on the other side lies a reality composed of the “forms of the given”, objects and relations of scientific experiment and observation. At this juncture, the “bridges” between the two spheres, of axiomatic scientific theories and “reality”, “are down”. But they are restored by the coordination of judgments, affirming relations among concepts, to facts, always concerning at least two objects and a relation between them.31 The target of Schlick's insistence upon this radical separation is clearly marked out; it is any version of synthetic a priori constitutive principles: “Thought never creates the relations of reality [Wirklichket]; it has no form which could imprint it, and reality allows no imprinting, for it is already formed”.32 Those of the neo‐Kantian school, Schlick declared, “commit the error of taking the conceptual wrapping for reality itself”.33

In Schlick's account, judgments, if not tautologies, explicit definitions, or false, are thus signs for facts of the world.34 Cognitive judgments, propositions representing new knowledge claims, are new combinations of old concepts occurring in other propositions; some of these concepts are previously known, for example, from an explicit definition that is based on a convention (“A yard is three feet”). In general, four different classes of possible judgments are distinguished:35 (1) definitions, a coordination completed through an arbitrary stipulation; (2) empirical judgments, designations of facts of experience; (3) hypotheses, judgments formed from known concepts for a provisional designation of facts, in the hope of attaining a univocal coordination; and (4) synthetic judgments a priori in the sense of Kant, noted above. Schlick will always deny the actual existence of the latter.

Of relevance here is the opposition between definitions or conventions and hypotheses. A reader of the first (1918) edition of the Allgemeine Erkenntnislehre might be inclined to think that there is little at issue in this distinction, since Schlick's view is that this only a relative difference, relative essentially to what is already known, that is, to the state of the system of scientific cognition at a given time. The more self‐contained and developed is the deductively connected scientific system of concepts and judgments, the more “genuine judgments” differ from definitions only in a “practical or psychological sense, not in a purely logical or epistemological one”.36 An axiomatized mathematical theory provides the illustrative example: it is to some extent arbitrary whether certain sentences are derived as theorems, or treated as axioms from whose consequences other judgments may be derived that ordinarily serve as definitions of the concepts. In the less deductively developed and self‐contained empirical sciences, the difference between definitions and “genuine judgments” appears to be clearer and better founded. For in the empirical sciences a definitional judgment first allots a given meaning to a concept, but then concepts designating real objects continually acquire “an ever richer content” through the process of inquiry and so the judgments containing them appear as instances of knowledge, “genuine judgments”, rather than definitions. But in principle the situation is no different; the difference in kinds of judgment is merely a relative one, and the same “linguistic formulation” may serve in either role:

Once a science has developed into a rounded‐out, more or less closed, structure, what is to count in its systematic exposition as definition and what as knowledge [Erkenntnis] is no longer determined by the accidental sequence of human (p.56) experiences. Rather one will accept as definitions those judgments that resolve a concept into such characteristics that one can construct from the same characteristics, many—perhaps even all—concepts of the given science in the simplest possible way.37

This fluidity between cognitive judgments and definitions is ultimately solidified through appeal to the principle of simplicity, in which, following Poincaré, Schlick located the operative criterion for conventional choice, even among judgments, or systems of judgments (theories) that can each be considered to be “true”. For if a judgment, formed within the interconnected scheme of concepts that is the axiomatized theory, designates a fact univocally, it is called “true”. Truth, the “only virtue” of judgments, is just the “univocal designation” of facts by judgments which means that different conceptual systems containing judgments univocally designating all the facts in question, may equally be deemed “true”.38 Underdetermination of theory by empirical evidence is expressly recognized; unlike contemporary scientific realists, Schlick regarded it as posing no particular obstacle for his entirely semiotic conception of truth.

It is in precise accord with this fluid distinction between hypothesis and definition in the first edition of Allgemeine Erkenntnislehre, that Schlick assessed the philosophical significance of the (special) principle of relativity in 1915. In this essay, published in December 1915, shortly after Einstein presented his generally covariant gravitational theory to the Prussian Academy on 25 November, Schlick allowed that both the Einstein “view” (without the ether) and the Lorentz “view” (with a substantial ether), can be reckoned as “true” because each gives a univocal designation of the kinematical facts of space and time measurement. A decision in favor of Einstein's theory of (special) relativity can be made only if one further accepts “the principle that the simplest theory, the one least encumbered with hypotheses, is to be regarded as a ‘true copy’ of reality”. Indeed, for Schlick in 1915, “the real” or “reality”, as that concept is presupposed in science, is to be defined through the principle of simplicity:

We can simply assert that among the possible assumptions the simplest should be designated as the one “corresponding to reality”. “Reality” is then just a word for that unknown reason which “brings it about” that certain theories yield the simplest type of natural regularity.39

Any assertion that nature is simple cannot be based on experience but must be a mere stipulation. Complex theories can always be thought up that equally provide a univocal designation of all the relevant facts. The prototype for this kind of approach to theory‐choice in the face of empirical underdeterminism, Schlick made quite clear, is Poincaré's conventionalist preference for Euclidean geometry.

Thus, although Schlick posed the issue to Cassirer in 1921 as “empiricist or Kantian”, it is very difficult to construe Schlick in 1915 as supporting anything like an empiricist interpretation of the geometry of space‐time. Indeed, there he subordinated empiricism to a conventionalism that nonetheless has realist aspirations, appealing to Poincaré's geometric conventionalism to secure his semiotic conception of truth as “univocal designation”, while tacitly criticizing Helmholtz, along with Gauss, for holding that an “empiricist” conception of physical geometry was possible: (p.57)

It is therefore no contradiction, but lies, rather, in the nature of the matter, that under certain conditions several theories may be true at once, in that they provide to be sure different, but still in each case completely univocal designation of the facts. One of them, indeed, will do this more skillfully and simply than all others, and one may therefore work with it alone, and even agree to call it the sole “correct” one, but a logically compelling reason for this may not at first be apparent. …  As [an] example we may refer to the possibility of using different geometries in the physical description of the world, without doing any harm to the univocality [Eindeutigkeit]. Henri Poincaré has shown with convincing clarity (although Gauss and Helmholtz still were of the opposing opinion), that no experience can compel us to lay down a particular geometrical system, such as the Euclidean, as a basis for depicting the physical lawfulness of the world. Rather, one can choose entirely different systems for this purpose, though in that case we also have at the same time to adopt other laws of nature. … We always measure, as it were, only the product of two factors, namely the spatial and the, in the narrower sense, physical, properties of bodies, and we can arbitrarily assume one of the two factors, so long as we merely take care that the product agrees with experience, which can be achieved by a suitable choice of the other factor. … The theory must now make it its task to so choose both factors, that the laws of nature are given the simplest possible expression. As soon as it succeeds in this, it appears to us with great persuasive power as the “correct” one. In the case of space, it is known that all experience teaches that it is by far the most convenient thing to base it on Euclidean geometry; physics can then be founded on the simplest assumptions of all (e.g., that a body retains its shape unaltered during a uniform translation]. We therefore absolutely [schlechthin] designate our space as Euclidean, although strictly speaking there is nothing that compels us to put nature's laws into Euclidean dress. That happens, as Poincaré expressed it, on the basis of a convention, and his view has therefore been given the name of conventionalism.40

With the general theory of relativity, Schlick understandably backed away from Poincaré's view that the simplest theory combining the two factors of geometrical and physical properties of bodies, will cast “nature's laws into Euclidean dress”. But the most noteworthy aspect of this passage is Schlick's unequivocal endorsement of geometric conventionalism, not empiricism.

Nor did a recognizably “empiricist” interpretation of physical geometry emerge in Schlick's 1917 monograph on the general theory of relativity, “Space and Time in Contemporary Physics”, although here, inexplicably, Helmholtz is now aligned with Poincaré, as a conventionalist. Once again, Schlick's relative distinction between hypotheses and definitions appears very clearly in his discussion of spatial measurement in the context of the general theory of relativity. There the reason guiding choice of the key stipulation—that certain bodies are to be regarded as rigid—is located in a so‐called Principle of Continuity, namely, that we “maintain continuity with the physics that has hitherto proved its worth”.

Comparing measuring rods and observing coincidences result in a measurement, as we have seen, only if they are founded on some idea, or some physical presupposition [Voraussetzung] or, rather, stipulation [Festsetzung]; the choice of which, strictly speaking, is essentially of an arbitrary nature, even if experience points so unmistakably to it as being the simplest that we do not waver in our selection.

(p.58) The “physical presupposition” or rather “stipulation” is that we are to regard the length of a rod as remaining constant, so long as its place, position, and velocity change only slightly. In other words,

we stipulate that, for infinitely small domains, and for systems of reference, in which the bodies under consideration possess no acceleration, the special theory of relativity holds. … The equations of the general theory of relativity must be, in the special case mentioned, transformed into those of the special theory.41

Now it is only in the absence of a firm distinction between physical “hypothesis” or “presupposition” and “definition” or “convention” that the validity of the special theory (in which Euclidean measure determinations are employed) in “infinitesimal regions” of the variably curved space‐times of the general theory, a claim often underwritten by the “infinitesimal principle of equivalence” criticized in chapter 2, §2.4.3, could be described as a stipulation. Such a characterization of the principle of equivalence can serve only to ease the assimilation of Schlick's 1917 treatment of geometry in the general theory of relativity to the conventionalism account he associated with Poincaré.

However, in the second (1925) edition of Allgemeine Erkenntnislehre, the fourfold classification of judgments of 1918 has become a threefold one.42 Hypotheses are no longer distinguished from other empirical judgments, but both are sharply differentiated from the broad class of definitions, analytic not synthetic judgments.43 “Concrete definitions”, the association of a name with a given particular object that can be “a quite arbitrary stipulation”, obviously belong here. But the class of definitions also contains conventions per se, definitions that enable a concept to apply to reality by attaining an univocal designation of the real (eine eindeutige Bezeichnung von Wirklichem).44 Spatiotemporal relations are “the true domain” of conventions, in particular, conventions asserting an equality of spatial or temporal intervals. In any case, they are quite different from “concrete definitions” and nontrivial in that it is “one of the most important tasks of natural philosophy” to investigate their nature and meaning.45 Yet in 1921, in his essay on Cassirer, Schlick still clearly recognized a relative distinction between definitions and hypotheses, and as discussed above, this relative distinction, together with its solidification through a principle of simplicity, is the core of his conventionalist account of truth as univocal correspondence. Why, then, did Schlick in 1925 efface this relative distinction between hypotheses and definitions?

Here is a clue, and it comes, most familiarly, from Quine: to the extent that a physical theory is regarded as confronting “the tribunal of experience” not statement by statement but only “here and there” through connections that implicate, sometimes in ambiguous fashion, large blocks of theory, to that extent do the individual concepts and judgments of the theory lack an individual empirical (or “cognitive”) meaning. And this is the case if there is only a relative, “practical or psychological, not a purely logical and epistemological” difference between judgments that are definitions and those that are “genuine”, empirical judgments designating, or purporting to designate, “facts. By the same token, the more indirect the empirical warrant of fundamental theoretical concepts may be allowed to be, the less suitable becomes any “empiricist” semantical analysis of the theory in question, given empiricism's mandate that observational evidence alone provides (p.59) the meaning of physical concepts, or can serve as sole justification for affirming or denying statements about the physical world. While Quine and others might still insist that a more liberal doctrine of empiricism can survive even these holist ties of theory to observation, Schlick in 1921 needed a more clearly identifiable variety of empiricism to counter the threat posed by Cassirer, who attempted to tether general relativity to critical idealism. Don Howard has recently put his finger on the difficulty here:

[T]he neo‐Kantian can argue that since the coordination does not occur empirical proposition by empirical proposition, since only whole theories are coordinated with reality, and since, therefore many different theories can equally well be coordinated with reality … which is to say that experience alone does not determine unambiguously our choice among possible theories, it is the function of synthetic a priori judgments to resolve the ambiguity.46

Then in order to save his purely designative conception of truth and to pose an empiricist alternative to the neo‐Kantians, Schlick needed to rein in the holist conception of physical theory as a system of statements whose central concepts are implicitly defined by the axioms of the theory. By fortuitous circumstance, the prototype of an empiricism suited to the theory of relativity had just appeared on the scene, promoted, no less, by Einstein himself. Or so Schlick apparently thought.

3.3 “Geometry and Experience”

At the Berlin Academy's Leibniz‐day public celebration on 27 January 1921, Einstein gave an address entitled “Geometry and Experience”. This short lecture, also issued separately in expanded form, would be regarded by logical empiricism as a paradigm‐defining text, fixing key parameters of logical empiricist philosophy of science. Prominently reprinted (in part) in the Feigl and Brodbeck reader in philosophy of science that virtually defined Anglo‐American philosophy of science when it appeared in 1953, Einstein's essay perhaps is best known for its clear statement of the distinction between the modern axiomatic conception of geometry (Einstein observed approvingly, in a manner “Schlick in his book on epistemology has thus very aptly characterized as ‘implicit definitions’ ”) and “practical geometry”, a contrast given a sharply terse formulation in the dictum: “In so far as the propositions of mathematics refer to reality [Wirklichkeit], they are not certain; and in so far as they are certain, they do not refer to reality”.47

Three strands of Einstein's argument readily stand out. First, he introduced and defended a conception there called “practical geometry” (“praktische Geometrie”), a geometry arising from the “empty conceptual schemata” of axiomatic geometry through a coordination of the latter to “practically rigid bodies”, that is, measuring rods and clocks that behave “as do solid bodies in Euclidean space of three dimensions”. The presupposition of such bodies, in turn, renders geometry an empirical science.

It is clear that the system of concepts of axiomatic geometry alone cannot make any assertion as to the behavior of those objects of reality which we designate as practically rigid bodies. To be able to make such assertions, geometry must be (p.60) stripped of its merely logical‐formal character by the coordination of experienceable objects of reality [erlebbare Gegenstände der Wirklichkeit] with the empty conceptual schemata of axiomatic geometry. To accomplish this, we need only add the proposition: fixed bodies are related, with respect to their possible situations, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclidean geometry contain assertions concerning the behavior of practically‐rigid bodies.

Geometry thus completed is evidently a natural science; we may regard it in fact as the most ancient branch of physics. … We would call the thus completed geometry “practical geometry” and distinguish it in the following from “pure axiomatic geometry”. … To this portrayed conception of geometry I attach special importance, because without it, it would have been impossible to set up the theory of relativity. Without it the following reflection would have been impossible: in a system of reference rotating relatively to an inertial system, the laws of situation of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz‐contraction; therefore in the admission of non‐inertial systems as equally justified systems, Euclidean geometry must be abandoned.48

With the assumption of “practically rigid bodies” (and “perfect clocks”), the metrical relations of space (space‐time) are not a matter of convention but can be empirically determined from measurements made with rods and clocks; in this way, a clear decision can be made regarding the Euclidean or non‐Euclidean character of physical space. Second, the position contrasted to that of “practical geometry” is a holist form of geometric conventionalism that Einstein apparently regarded as stemming from Poincaré's rejection of the concept of actually rigid bodies. But with this denial, Einstein remarked, practical geometry's “original, immediate relation between geometry and physical reality [Wirklichkeit] is destroyed”.49 Third, Einstein nonetheless admitted, against his “practical geometry”, that the position of Poincaré is, sub specie aeterni, in principle correct. These different aspects of Einstein's argument shall now be scrutinized to reveal the dialectical interplay between them.

The ostensible main point of Einstein's lecture is the argument that the metric of the space‐time continuum is empirically determinable (and non‐Euclidean) against a conventionalist view of geometry of physical space identified with Poincaré. One might well wonder why Einstein believed it necessary to uphold the viewpoint of the empirical determinability of the geometry of space‐time at this time, that is, just a little over a year since the results of the British expedition confirming Einstein's theory of gravitation were announced (on 6 November 1919) from the podium of the Royal Society of London. One reason is certainly a concern to portray the historical development of the theory, which is alluded to at the end of the above quotation. As John Stachel has shown in detail, Einstein tacitly refers here to the case of a uniformly rotating disk, the simplest example of a stationary gravitational field.50 In particular, the assumption that rigid measuring rods correspond to distances on the rotating disk enabled Einstein to conclude (in 1912) that, due to the Lorentz contraction of the rods placed upon the circumference of the disk, the geometry of the disk could not be Euclidean, a crucial step down the road to the curved space‐times of general relativity.

(p.61) But a more general reason is this: although fully aware that the concept of a rigid body is not really permissible in his gravitational theory of variably curved space‐time, Einstein considered the empirical basis of his theory as lying in the connection of the line element ds to the rod and clock measurements of “practical geometry”. In particular, Einstein used the supposition that the “segment” or rather “tract” (Strecke) between two neighboring points of space was empirically definable independently of the theory by the extension of an “infinitesimal” rigid rod connecting them, “normed” as a unit interval.51 Similarly for timelike curves, a unit of duration was normed by considering the periods of two ideal clocks, running always at the same rate, no matter when and where they are brought together and locally compared. Then distances, angles, and durations in the theory could be presumed to be read off directly from the measurements of rods and clocks, paving the way to the British Expeditions' observations confirming the theory's successful prediction of the “bending” of light rays passing through the strong regions of the solar gravitational field. The idealization of rigid rods and regular clocks—that is, measuring appliances reckoned as unaffected by the presence of a surrounding gravitational field or other fields—served as a “bridge” to link the phenomena of gravitational mechanics to Einstein's non‐Euclidean Riemannian geometry of curved space‐time. For example, assuming “practical geometry”, the metrical “distance” between finitely separated points P and Q corresponds to the measure obtained by the number of times an “infinitesimal” rod could be laid down along a “straight line” (as given by a light ray) joining the two points. On the other hand, the “distance” can be theoretically computed by integrating the invariant interval ds 2 between all neighboring points P′, P″, … Q along a path connecting P and Q. The intervals ds 2 are found from the components of the metric tensors that are functions of the coordinates of the respective points. In turn, these components can be calculated from the Einstein field equations and so depend on the amount of “matter” (momentum, energy, stress, etc.) surrounding the region containing P and Q. According to the principle of general covariance, the coordinate patch covering the region of the two points need not be Cartesian or Galilean (i.e., a rigid grid); no metrical significance is immediately attributable to coordinate differences. But in the supposition that the measured and the theoretical value for the “distance” between the two points are the same, Einstein located the empirical basis of his gravitational theory and so rendered its non‐Euclidean geometry a part of physics.

However, as discussed in chapter 4, since the spring of 1918 Einstein had continually inveighed, in private and in public, against Hermann Weyl's theory of gravitation and electromagnetism, on the grounds that it doesn't permit rods and clocks to exhibit the behavior that makes them the suitable instruments of measurement that they, in fact, are. In that theory, rods and clocks (or rather the radii of atoms and their spectral frequencies of vibration) are not independent of their position in space and time but rather depended, Einstein argued, on their “prehistory”, that is, the electromagnetic fields through which they had passed. Thus, two atoms of, say hydrogen, should display different spectra if one, but not the other, had passed through a strong electromagnetic field. For the time being, I leave aside whether this is an adequate rebuttal of Weyl's theory; what is (p.62) important here is that Weyl's theory lacked the direct connection to experience as did general relativity via the above‐mentioned “norming”. Moreover, to Einstein, Weyl's theory wrongly predicted that the spectral frequencies of the chemical elements should not be constant and independent of position as in fact they are observed to be.

Although Einstein's objection and its subsequent elaboration by Pauli (1921) persuaded nearly all interested parties, Einstein felt it incumbent upon him to raise the cudgel once again against Weyl in this Berlin Academy lecture of January 1921. Without mentioning Weyl's theory by name, Einstein reiterated his principal objection, that the demonstrable existence of sharp spectral lines of the atoms of the chemical elements, no matter what the prehistory of the atom had been (i.e., no matter what electromagnetic fields it had passed through), provided “compelling empirical proof” (überzeugenden Erfahrungsbeweis) for the “basic postulate of practical geometry”, that is, of the existence of infinitesimally rigid rods and clocks. Indeed, it is this assumption, Einstein continued, on which ultimately rests the physical meaningfulness at all of speaking of a metric in Riemann's sense within the four‐dimensional space‐time continuum.52 Thus, Einstein argued that the very applicability of Riemannian geometry to the physical world presupposed the existence of infinitesimal rigid rods and ideal clocks, and so the possibility of the empirical confirmation of the general theory of relativity rested upon the supposition that these idealized bodies give physical meaning to the concepts “unit measuring rod” and “unit clock (period)”. Measurements with these instruments physically attest to the metric field of gravitation. This posited direct connection with experience was, to Pauli, “the most beautiful achievement of the theory of relativity”, even though “logically, or epistemologically, this postulate does not admit of proof”.53

To “practical geometry's” conception of the direct linkage of the Riemannian geometry of general relativity to experience, Einstein opposed what he termed the view of Poincaré that it is a matter of convention which geometry we take to obtain in physical space. Recall that Poincaré had argued that in the absence of truly rigid bodies, purely geometrical statements affirm nothing about experience until they are combined with statements of physics.54 But this meant that in answer to the question as to the nature of the geometry of physical space, any geometry can be chosen since thereby one commits oneself only to a set of ideal propositions, as long as the supposed laws of the behavior of physical objects can be adjusted to be in agreement with what is actually observed. According to Einstein, this view is concisely represented in the formulation

Total theory = G + P

In other words, a geometry G can be chosen arbitrarily and also part of the system of physical laws P, as long as the remainder of P enables the total theory to be brought into agreement with experience. As just shown, Schlick both formulated and endorsed this holist and conventionalist conception of physical geometry in his writings on relativity theory prior to 1921. Moreover, for Schlick and for Einstein's Poincaré sub specie aeterni, it is not really germane that the actual Poincaré additionally thought that Euclidean geometry would always be adopted as the simplest geometry, with the corresponding adjustments to physical laws (e.g., (p.63) maintaining that light rays no longer traversed Euclidean straight lines). But even while arguing against geometric conventionalism, Einstein conceded Poincaré's point that there are, in fact, no “actually rigid bodies”; indeed, Poincaré had maintained that this is a confused conception, a kind of categorical mistake. The concepts of the fixed body and clock, Einstein admitted, “do not play the role of irreducible elements in the conceptual edifice of physics but rather of composite structures which should play no independent role in the construction of theoretical physics”.55 What Einstein meant is that general relativity, which in principle is capable of encompassing all matter fields into the geometry of space‐time (in 1921, this was only electromagnetism), should, again in principle, explain material structures (e.g., rods and clocks) as composite structures whose structure and behavior are derivable from the theory's field equations. Hence, connecting such a theory to experience by means of concepts treated as independent of the theory (in particular in the assumption of rigidity), when in fact they are not, is a less than consistent procedure.

This was not really a new concession by Einstein. A year before completion of the general theory of relativity in November 1915, Einstein had expressed views on the connection of geometry to experience in a manner similar to that which he designated in 1921 as “practical geometry”. However, in that context, his attention was limited to pointing out that the advent of field theories, with their prohibition of action‐at‐a‐distance, presented a new and critical perspective on Euclidean geometry.

Before Maxwell, the laws of nature were, in spatial relation, in principle integral laws; this is to say that distances between points finitely separated from one another appeared in the elementary laws. This description of nature is grounded upon Euclidean geometry. The latter signifies at first nothing other than the system of consequences of the geometrical axioms; they have, in this respect, no physical content. However, geometry becomes a physical science by adding the requirement that two points of a “rigid” body must be separated by a determinate distance, independent of the position of the body. Propositions supplemented through this stipulation [Festsetzung] are (in the physical sense) either applicable [zutreffend] or inapplicable. In this extended sense, geometry forms the foundation of physics. From this viewpoint, the propositions of geometry are to be considered as integral physical laws, since they deal with distances of finitely separated points. … Through and since Maxwell, physics has undergone a through‐going radical change in gradually carrying through the demand that distances of finitely separated points may no longer appear in the elementary laws, that is, “action at a distance theories” [Fernwirkungs‐Theorien] are replaced by “local‐action theories” [Nahewirkungs‐Theorien]. In this process it was forgotten that also Euclidean geometry—as employed in physics—consists of physical propositions that from a physical viewpoint are to be set precisely on the side of the integral laws of the Newtonian point mechanics. In my opinion, this signifies an inconsistency from which we should free ourselves.56

Einstein went on to introduce his readers to the novel idea that his gravitational theory, in treating coordinates as arbitrary parameters in the space‐time continuum, reduced the integral laws of Euclidean geometry to differential laws, removing the mentioned inconsistency, and appearing as a natural extension of the (p.64) requirement of a “local‐action theory”. But if set against the 1921 essay “Geometry and Experience”, what is particularly striking about this 1914 passage is the lack of a contrast to the method of (what is later termed) “practical geometry” or any anticipation of another tie of geometry to experience other than through a stipulation about the invariant distance between points on a “rigid” body. In other words Einstein did not recognize in 1914, as he did in 1921, Weyl's point (see further in chapters 4 and 6) that reliance upon the rigid bodies of geometrical measurement is itself an inconsistency with the character of the general theory of relativity as a “local action theory” (Nahewirkungstheorie) but represented a remnant of the Ferngeometrische past of Euclidean geometry.57 In 1921, Einstein additionally admitted that, in principle, the “practically rigid” behavior of such complicated physical structures as rods and clocks (manifesting what Weyl would call the “natural gauge of the world”; see chapter 6, § should be explicable as a remote consequence of the field equations. In a theory such as general relativity, such empirically direct ties of geometry to experience via “practical geometry” are unexplained explainers. This, and not a conventionalist freedom to always choose Euclidean geometry, is the sense in which Poincaré is right sub specie aeterni. Nonetheless, Einstein stated his belief that at the present stage of knowledge, that is, in the absence of a field theoretic account of matter, these concepts must be provisionally accepted.

Einstein's admission, in 1921, of the sub specie aeterni correctness of Poincaré's point of view is not a concession to conventionalism, that is, to freedom to choose any geometry we like, but to the inevitable epistemological holism of a theory in principle capable of explaining its own measuring applicances, and so its ties to observation. Poincaré's position is valued for its principled unwillingness to consider certain physical objects as “geometrical”, that is, as ideal, and so as independent from the field laws that, in principle, are accountable for the behavior of all material structures. But this is precisely the point of view of what Einstein later termed “a consistent field theory”. It is accordingly important to keep sight of the qualified character of Einstein's methodological analysis in favor of “practical geometry”. By 1921, Weyl had shown that there are other, less direct means of connecting the geometry of space‐time to observation through the paths of freely falling “test particles” and of light rays, avoiding the inconsistent assumption of even infinitesimal rigid bodies in a theory of the gravitational and, perhaps, electromagnetic fields. In such a procedure, the tie of the total theory, geometry plus physics, to experience involves quite intricate and theory‐internal complexions: tracks of force‐free neutral test particles of negligible mass are taken to manifest geodesics of the affine structure of space‐time; the equation of motion of such particles is itself (at least in the case of a pure gravitational field) derivable from the theory's field equations. The totality of the affine geodesics endows the space‐time manifold with a projective structure. Paths of light rays, in turn, provide the space‐time manifold with a causal‐conformal structure; it can then be shown that the paths of light rays (conformal geodesics) are a limit case of the projective structure and that together they give enough information to determine a metric at a given point. (The projective and the conformal structure determine the metric, up to a factor of scale; see chapters 4 and 6.) This construction of the metric of space‐time shows that it is neither necessary nor desirable to posit (p.65) a theory‐independent (i.e., of gravitational theory) definition of the distance between two neighboring points.58 Such a construction, to be sure, involves its own assumptions, but these are assumptions compatible with the field equations of the theory. In so many words, without the assumption of rigid bodies, the empirical foundation of general relativity acquires the epistemological and semantic complexity that fall under the rubric of holism.

It is therefore worth noting that within just a few years Einstein took repeated pains to distance himself from an unqualified commitment to “practical geometry” on grounds that are explicitly holist in nature. Here just two instances are cited. First, in the prominent context of his “Nobel lecture” (1923e), delivered to the Nordic Assembly of Naturalists at Gothenburg (serving in lieu of a Nobel prize lecture), he brought up the “deficiency of method” in the stipulation that measurement bodies are rigid. Before doing so, Einstein stated the kind of empiricist meaning criterion often deemed characteristic of the method of the theory of relativity:

[C]oncepts and distinctions are only admissible to the extent that observable facts can be assigned to them without ambiguity (stipulation that concepts and distinctions should have meaning). This postulate, pertaining to epistemology, proves to be of fundamental importance.

A bit further on, however, Einstein returned to consider this empiricist “meaning stipulation” in regard to the notions of rigid body and uniform clock of chronogeometrical measurement:

The concept of the rigid body (and that of the clock) has a key bearing on the foregoing consideration of the fundamentals of mechanics, a bearing which there is some justification for challenging. The rigid body is only approximately achieved in nature, not even with desired approximation; this concept does not therefore strictly satisfy the “stipulation of meaning”. It is also logically unjustifiable to base all physical consideration on the rigid or solid body and then finally reconstruct that body atomically by means of elementary physical laws which in turn have been determined by means of the rigid measuring body. I am mentioning these deficiencies of method because in the same sense they are also a feature of the relativity theory in the schematic exposition which I am advocating here. Certainly it would be logically more correct to begin with the whole of the laws and to apply the “stipulation of meaning” to this whole first, that is, to put the unambiguous relation to the world of experience last instead of already fulfilling it in an imperfect form for an artificially isolated part, namely, the space‐time metric. We are not, however, sufficiently advanced in our knowledge of nature's elementary laws to adopt this more perfect method without going out of our depth. At the close of our considerations we shall see that in the most recent studies there is an attempt, based on ideas by Levi‐Civita, Weyl, and Eddington, to implement that logically purer method.59

Eddington was the first to build up a “world geometry” of the curved space‐time continuum by beginning with an affine connection, making the metric of secondary fundamental importance (see chapter 8). Obviously, in such a theory (in 1923 envisaged as a field theory of gravitation and electromagnetism), originally nonmetrical, there can be no immediate connection—requiring both the supposition of (p.66) a metric and a “realization” of distance through rigid rods—of metrical notions with experience.

Another, even more explicit, methodological endorsement of holism in the ties of geometry to experience is given the essay “Non‐Euclidean Geometry and Physics” (1925a). Although Einstein's main thesis in this article is that the geometry of space‐time in general relativity is empirically determined to be non‐Euclidean, the methodological issues between “practical geometry” and holism are posed very clearly.

According to [the] more refined conception of the nature of the fixed body and of light, there are no natural objects which correspond exactly in their properties to the basic concepts of Euclidean geometry. The fixed body is not rigid [starr], and the light ray does not rigorously embody the straight line; of course in general, it is not a one‐dimensional structure. According to modern science, geometry by itself [allein] anyway corresponds to no experiences, but rather only geometry together with mechanics, optics, and so on.

A few paragraphs later Einstein returns to the issue of the tie of geometry to experience, remarking that “one must take up either one of two consistent standpoints”. The first is the standpoint of the “practical physicist”:

Either one accepts that the “body” of geometry in principle is actualized through the fixed body of nature, if only certain regulations are imposed regarding temperature, mechanical demands, and so on. Then, to the “tract” [“Strecke”] of geometry corresponds a natural object, and with this all propositions of geometry attain the character of expressions about real bodies. This standpoint was represented especially clearly by Helmholtz, and one can add that without it, the setting up of the theory of relativity would have been practically impossible.

The other standpoint, as noted above, is however that of “modern science”:

Or, one in principle denies the existence of objects which correspond to the basic concepts of geometry. Then geometry by itself contains no expressions concerning objects of reality [Wirklichkeit] but only geometry together with physics. This standpoint, which may be more perfect [vollkommenere] for the systematic representation of a completed physics, was represented especially clearly by Poincaré. From this standpoint the total content [Inhalt] of geometry is conventional; which geometry is preferred depends upon, through its use, how “simple” a physics can be set up that is in agreement with experience.60

Just as in “Geometry and Experience”, Einstein then remarks that he chooses the first standpoint, that of the “practical physicist” and so, of Helmholtz, “as better agreeing with the current position of our knowledge”. If this is done, the issue between the Euclidean or non‐Euclidean character of the geometry of space‐time is plainly posed and, Einstein argued, clearly answerable. The other standpoint, of “modern science”, is that of Poincaré and holism, with the overall simplicity of physics together with geometry as the determining criterion in the choice of a particular geometry. But even from this standpoint, although unlike in Poincaré, choice of Euclidean geometry within “modern science” is really nonadmissible. This point is tacitly conceded at the end of the lecture in referring to the further generalizations beyond the Riemannian geometry of general relativity by Weyl (p.67) and Eddington, based upon Levi‐Civitá's concept of the “infinitesimal parallel displacement” of a vector (see chapters 4, 6, and 8). These theories have shown, Einstein concluded, that “the ideas which have developed out of non‐Euclidean geometry have proven eminently fruitful in modern theoretical physics”.61 Such a fruitfulness of geometric ideas in physics would seem to belie Einstein's claim that from the standpoint of “modern science”, “the total content of geometry is conventional”. Moreover, in these theories, the question of the geometry of space‐time within the total system comprised of geometry plus physics centers on a choice among different non‐Euclidean geometries.

In sum, the holism of the standpoint that Einstein saw Poincaré as representing “especially clearly”, is independent of Poincaré's own conventionalist choice of Euclidean geometry. Einstein's conjunction of the distinct issues of holism and conventionalism is understandable, rooted as it is in Poincaré's own discussions of geometric conventionalism. But the dynamical character of the space‐time metric of general relativity provides ample grounds for disentangling epistemological holism from conventionalism. That geometry could be dynamical was a possibility not taken seriously by Poincaré in the light of his remark that Riemann's geometries of variable curvature, which are “incompatible with the motion of a rigid figure”, “could never therefore be other than purely analytic”.62 But in any case, these two texts, of 1923 and 1925, suffice to show that Einstein's account of the connection of space‐time geometry to experience via the expedient of “practical geometry” was considerably more nuanced and provisional than Schlick, and subsequently Reichenbach, would take it to be. Far from simply choosing a stipulation through which a metrical concept acquires physical meaning, the issue of the empirical character of geometry can be assessed from two different and complementary standpoints, either of which has distinct advantages. “Helmholtz” and “practical geometry” had proved invaluable in the heuristic genesis and initial confirmation of general relativity and, moreover, provided a lens through which to focus attention on the speculative and unphysical character of Weyl's theory. Yet “Poincaré” and “completed physics” (and, although unstated, “Weyl” and “Eddington”) are necessarily the perspective of a “unified field theory” such as Einstein, after 1921, sought with unrelenting determination. Within unified field theory, the standpoint of epistemological holism in linking space‐time geometry with experience is no longer merely an option.

3.4 Helmholtz and “Schlick's Helmholtz”

In 1921, Schlick and Paul Hertz, a physicist in Göttingen and relative of the famous Heinrich Hertz, editorially collaborated to publish a centenary collection of Helmholtz's Epistemological Writings annotated with extensive “elucidatory” footnotes.63 Significantly, this collection of four papers included Helmholtz's two classic articles on the foundations of geometry, “On the Facts Underlying Geometry” (“Über die Tatsachen, die der Geometrie zugrunde liegen”, 1868) and “On the Origin and Significance of the Geometrical Axioms” (“Über den Ursprung und die Bedeutung der geometrischen Aziome”, 1870), as well as “the free and untrammeled statement of his philosophical position”,64 “The Facts in Perception” (“Die Tatsachen in der (p.68) Wahrnehmung”, 1878). Schlick wrote the elucidatory notes to the 1870 and 1878 papers, and it is here that the immediate impact of Einstein's lecture shall be discerned.

Helmholtz published “On the Facts Underlying Geometry” in 1868 in the Nachrichten of the Göttingen Academy of Sciences; the previous year, 1867, Riemann's celebrated Habilitationsrede of 1854 “On the Hypotheses Underlying Geometry” had been published after Riemann's death by Dedekind in the Abhandlungen of that Academy. Indeed, the publication of Riemann's essay came as a revelation to Helmholtz, who, in 1866, had taken the pre‐Riemannian position that free mobility of rigid bodies implied the validity of the Euclidean axioms.65 But as still appears from the echo in his chosen title, Helmholtz's 1868 essay is ostensibly an attempt to replace the “hypotheses” Riemann saw as underlying geometry with “facts”; it will emerge, however, that Helmholtz's essay treats only a considerably more restricted conception of geometry. At the very beginning, Helmholtz posed the fundamental question he would attempt to answer: to determine to what extent the propositions of physical geometry have “an objectively valid meaning” and to what extent are they dependent on definitions or the form of descriptions. Despite his subsequent reputation as a “geometric empiricist”, Helmholtz immediately continued that in his opinion “this question is not to be answered all that simply”. The principal difficulty facing an empiricist account of geometry is then squarely pinpointed:

[I]n geometry we continuously deal with ideal structures [idealen Gebilden] whose bodily representation in actuality [Wirklichkeit] is always only an approximation to requirements of the concept, and we only decide whether a body is rigid [fest], whether its surfaces flat, its edges straight, by means of the same propositions whose factual correctness the examination is supposed to demonstrate.66

The difficulty is that the concept or idea of a rigid body must be already be operatively legitimate in order to be in a position to know whether any given body is approximately rigid. For this reason, the concept of a perfectly rigid body is not itself acquired from experience. Nonetheless, it must be presupposed in the practice of geometric measurement, particular instances of which serve to confirm or disconfirm whether geometrical‐physical space has a Euclidean structure.

In this 1868 paper, Helmholtz sought to derive Riemann's central hypothesis, that the length dl of an infinitesimal Pythagorean line segment is expressed by a quadratic function of the coordinate differentials (written here in the modern way),

d l 2 = g i k d x i d x k

from certain hypotheses that expressed “facts” primarily about the observable behavior of the standard bodies of geometrical measurement, in particular, the rigid measuring rod. As he made clear at the outset, Helmholtz's concern, unlike Riemann's more general investigation, was limited to the consideration of “actual space” (wirklicher Raum), a space satisfying the requirement that in it, finite systems of fixed points (rigid bodies) could move around without distortion. With the constraint imposed by the free mobility of finite fixed bodies, he could show that Riemann's hypothesis was derivable from four less restricted assumptions.

(p.69) Helmholtz's characterization of actual space as allowing free mobility of rigid bodies stemmed from his supposition that all geometrical measurement, and so the very possibility of physical geometry, is based on the observation of the relation of congruence between spatial magnitudes. Observations of congruence between bodies, according to Helmholtz, presupposed the possibility of motions of fixed finite bodies up to and adjacent to one another, as well as that congruence of spatial magnitudes is independent of all motions. So the facts about congruent spatial magnitudes are facts about rigid bodies, and their motions, a considerably different view of measurement from that of Riemann, for whom geometric measurement presupposed only the total or partial superposition of one ideally thin and perfectly flexible “measure string” or thread on another.67 These facts are expressed in the last three of the four hypotheses underlying Helmholtz's investigation; the first is taken directly from Riemann. They are 1) that a space of n‐dimensions is an n‐fold extended manifold (meaning a point in it is specified by n independent and continuously varying coordinates), and the motion of a point in such a manifold is accompanied by a continuous change in at least one of the coordinates; 2) that there exist mobile finite rigid bodies, or fixed point systems; 3) that there is no constraint on these motions, but completely free mobility of these bodies; and finally, 4) that space is monodromous (i.e., that two congruent bodies are still congruent after one of them has undergone a complete rotation about any axis of rotation); Sophus Lie, using the language of continuous (Lie) groups, later showed this hypothesis (for spaces of three or more dimensions) to be redundant.68 As clarified by Lie, justification for Riemann's hypothesis of an “infinitesimal Pythagorean metric” on an n‐dimensional manifold involved showing that the isometries of this metric (its congruences, or equality of measures) are precisely captured by an infinite parameter (continuous) group of motions acting on the manifold such that, for each motion carrying a rigid body from any given point P to any other point Q, there is exactly one transformation in the group.

One immediate difference between Helmholtz and Reimann that for Riemann, free mobility is not assumed for finite bodies but only for infinitely small bodies, corresponding to a Riemannian manifold's assumption of “flatness in its smallest parts”. There are two further restricting conditions: (a) unlike Riemann, Helmholtz was concerned only with the case of a three‐dimensional manifold, corresponding to actual space, and (b) since the aim is to drive Riemann's result, expressed in terms of differentials of the coordinates, Helmholtz considered “only points having infinitely small differences in the coordinates”. So a “congruence independent of limits will be presupposed only for infinitely small spatial elements”,69 ruling out applicability of his hypotheses to bodies of arbitrary size. As thus pertaining only to infinitesimal displacements rather than finite motions, Helmholtz apparently believed that his hypotheses were weaker than Riemann's axioms. In fact, they are completely different; Lie showed that thereby they have been radically changed since it is then unclear what Helmholtz's axioms, as interpreted infinitesimally, assert about observable finite motions and, conversely, what the finitely interpreted axioms state about his infinitesimal displacements, since neither the finite nor the infinitesimal cases are inferable from one another.70 Moreover, precisely because Helmholtz's derivation pertains only to infinitesimal displacements, it does not furnish what Helmholtz claims for it (already in the title of his paper) as its (p.70) significance, for pending some physical understanding of the extent of the spatially “infinitesimal”, such as relativity theory later provided, the validity of his axioms in the infinitesimal cannot be connected, except at most indirectly and hypothetically, with the “observable facts” about the free mobility of rigid finite bodies, that is, as an integral and not a differential law. Helmholtz's proof therefore does not, and cannot, accomplish what it claims to do, a derivation of Riemann's hypothesis from observable facts. This remarkable lapse is alluded to briefly in notes of P. Hertz to Helmholtz's 1868 paper.71 But with the exception of the second, there is no reason to linger over the other hypotheses. So consider the salient issues Schlick raised with Helmholtz's hypothesis about the existence of rigid bodies.

The second hypothesis notably requires a definition of rigid body, but to all appearances the definition given is circular, for it invoked a conception of points already fixed in space. The definition, cited from the 1870 geometrical paper, reads:

[T]the definition of a rigid body can now only be given by the following characteristic: Between the coordinates of any two points belonging to a rigid body, an equation must exist that expresses an unchanged spatial relation between the two points (which finally turns out to be their separation) for any motion of the body, and one which is the same for congruent point pairs. Such point parts however are congruent, which can successively coincide with the same fixed point pairs in space.72

The apparent circularity was noted by both P. Hertz and by M. Schlick in their respective “elucidations” to Helmholtz's two papers on the foundations of geometry reprinted in the centenary collection. Schlick's comment to the above passage (n. 31 to the text) is worth quoting at length:

This definition reduces congruence (the equality of two tracts [Strecken]) to the coincidence of point pairs in rigid bodies “with the same fixed point pairs in space” and thus presupposes that “points in space” can be distinguished and held fixed. This presupposition was explicitly made by Helmholtz … , but for this he had to presuppose in turn the existence of “certain spatial structures which are regarded as unchangeable and rigid”. Unalterability and rigidity … cannot for its own part again be specified with the help of that definition of congruence, for one would otherwise clearly go round in a circle. For this reason the definition seems not to be logically satisfactory.

One escapes the circle only by stipulating by convention that certain bodies are to be regarded as rigid, and one chooses these bodies such that the choice leads to a simplest possible system of describing nature [here Schlick refers to Poincaré, The Value of Science, p. 45 of the German edition]. It is easy to find bodies which (if temperature effects and other influences are excluded) fulfil this ideal sufficiently closely in practice. Then congruence can be defined unobjectionably (as by Einstein in “Geometrie und Erfahrung”, p. 9] as follows: “We would call a tract [Strecke] the embodiment set out by two marks on a practically rigid body. We imagine two practically rigid bodies with a tract marked on each. These two tracts shall be called ‘equal to each other’ if the marks on the one can constantly be brought into coincidence with the marks on the other”.73

A noticeable symptom of this alleged difficulty with Helmholtz's definition is that, although he appeared to recognize that the notion of a rigid body is an (p.71) idealization (as in the remarks at the very beginning of his 1868 paper), in point of fact his text several times gives evidence of a stubborn belief in the existence of “actually rigid bodies”. Now, as highlighted above, Helmholtz clearly thought that the axioms of geometry also make assertions about the behavior of physical measuring bodies. But he also indicated that he considered it meaningful to assert of two bodies once held to be congruent that they might, at some later time, and perhaps other situation, “have changed in the same manner” although they still are observably congruent. The reason is that Helmholtz, while recognizing the intertwining of geometry and mechanics in the concept of congruence, consistently maintained that the notion of a rigid body is constitutive of the concept of congruence on which geometrical measurement rests. This was expressly indicated earlier in his paper, where he noted,

So all our geometrical measurements rest upon the presupposition that the measuring instruments which we take to be rigid, actually [wirklich] are bodies of unchanging form.74

Schlick inserted a footnote at the word “actually” that reads:

In the little word “actually” there lurks the most essential philosophical problem of the whole lecture. What kind of sense is there in saying of a body that it is actually rigid? According to Helmholtz's definition of a rigid body … , this would presuppose that one could speak of the distance between points “of space’ ” without regard to bodies; but it is beyond doubt that without such bodies one cannot ascertain and measure the distance in any way. Thus one gets into the difficulties already described in note 31. If the content of the concept “actually” is to be such that it can be empirically tested and ascertained, then there remains only the expedient already mentioned in that note: to declare those bodies to be “rigid” which, when used as measuring rods, lead to the simplest physics. Those are precisely the bodies which satisfy the condition adduced by Einstein (compare note 31). Thus what has to count as “actually” rigid is then not determined by a logical necessity of thought or intuition, but by a convention, a definition.75

In fact, this “correction” completely transforms Helmholtz's views on space, as I will show. But according to Schlick, Helmholtz, either unawares or disingenuously, in seeking to base geometry upon “facts” about “actually rigid bodies”, has given a circular definition of such bodies. The solution, in the light of Schlick's understanding of Einstein's “Geometry and Experience” is to break the circularity by a stipulation that certain bodies are “rigid” when, using those bodies as measuring rods, the “simplest physics” results. In this way, truths concerning geometrical‐physical space are part of such of system of “simplest physics” and are determined by measurements made with the fiduciary rigid bodies. The geometry that enables such a system of physics is then empirically ascertainable.

However, if Helmholtz's geometrical papers are set in the context, as he himself did, of his prior “investigations on spatial intuitions (räumlichen Anschauungen) in the visual field”, then another, and more internally consistent, reading of this seemingly circular definition of rigidity emerges.76 In a recent examination of the relationship between Schlick's semiotic epistemology, epitomized above in §3.2a, and Helmholtz's “theory of signs” (Zeichentheorie), Michael Friedman has described how Helmholtz's researches in the psychology and physiology of perception lay (p.72) behind the most elaborate statement of his epistemology, “On the Facts in Perception” (1878), and its principal conclusion, which is critical of the causal realist theories of perception.77 According to his “theory of signs”, perceptions, our sensory representations of objects external to our body, are symbols or signs for these objects that need not resemble them in any way (indeed, the idea of a comparison is not even thinkable). For Helmholtz, the external cause standing behind the play of our sensations is not an object in another realm existing behind the veil of our perceptions but is just the lawlike relation governing the patterns of sensations themselves. As Friedman documents, Helmholtz's investigations of geometry were a key step in his rejection of the causal realist view of perception he had previously entertained. In particular, his researches on visual perception in the late 1850s and early 1860s had led to the conclusion that our ability to localize things in space stem from an innate capacity to imaginatively construct, from given sense impressions of an object or objects, lawlike sequences of the sense impressions that would or could be obtained through the voluntary movement of our bodies toward them, away from them, around them, and so on. Subsequently, in his researches on geometry, Helmholtz saw that these patterns of lawlike sequences of sensations, connected with the actual and possible motions of our bodies, provide the means for a representation of space itself.78 In this sense, space is not some substantive arena in which objects behind the veil of perception are sporting about, but rather, through the anticipated possible motions of our bodies,

a given form of intuition, possessed prior to all experience, in so far as its perception is connected with the possibility of motor impulses of the will for which the mental and corporeal capacity had to be given us, by our organization, before we could have spatial intuition.79

That we have such a form of “outer intuition” for spatial relationships is a priori in the sense that its origin lies the physiological and psychological makeup that affords the possibility of an anticipation of the patterns of sensations stemming from imaginatively projected voluntary motor impulses of our bodies; the perception of the space of intuition is thus bound up with the possibility of the volitional motor impulses. In turn, a mathematical representation of this form of “outer intuition” can be constructed from the possible lawlike sequences of sensations that stem or would stem from an imaginative free mobility of fixed bodies throughout the space. Such a conception of space is the actual space of physical objects, for the concept of congruence, underlying the possibility of geometrical measurement, itself rests on possibility of free mobility of rigid bodies of measurement, and judgments of congruence are based on the lawlike uniformity in sensations in the perception of superimposed bodies.

Contrary to Kant, this space of intuition is not necessarily described by the axioms of Euclidean geometry; all that follows from the condition of free mobility of fixed bodies is that this space has the mathematical structure of a three‐dimensional space of constant curvature. If it is further supposed that this space is infinite, that two bodies can be continuously moved indefinitely far apart, then this space can be either a hyperbolic space of constant negative curvature or Euclidean flat space. In either case, the particular axioms that describe geometrical–physical space are only to be discovered from the spatial measurements made with rigid (p.73) measurement bodies. Different systems of axioms of geometry (Euclidean, elliptic, hyperbolic) are compatible with the a priori form of spatiality that is mathematically describable as a space of constant curvature, but it is an empirical discovery which set of axioms accurately characterize geometrical figures in this space. This relation between space as an a priori form of outer intuition and the axioms of geometry is summarized in Helmholtz's well‐known aphorism, “Space can be transcendental without the axioms being so”.80

It is now possible to see how Helmholtz could have given the definition of “rigid body” that in fact he did. Space as a form of “outer intuition”, spatiality as such has the structure of a space of constant curvature, as it must if the free mobility of fixed bodies is possible. In such a space, systems of points can be represented as rigid in supposing an ideal extension between any pair of such points, ideal as not occupied by any material body. For purposes of mathematical representation, coordinates can be assigned to the points in such a way that to any difference in coordinates between pairs of points, there corresponds an ideal dematerialized fixed extension. Geometrical measurement is then possible on the presupposition that the measurement bodies of geometry are actually bodies with such a constant fixed extension. It is this meaning‐constituting presupposition, according to Helmholtz, permits speaking of “actually rigid bodies”.81 Regarded as a condition of the possibility of geometrical measurement at all, such measurements being restricted to a space of three dimensions in which free mobility of fixed bodies can be intuitively represented, it is not a conventional stipulation that some bodies are to be considered rigid nor is it a naive confusion regarding actual, physical bodies that, as Helmholtz knew very well, are never more than approximately rigid.

The Helmholtzian view of physical geometry is thereby distinct from a “geometric empiricism” holding that the metrical relations of space can be straightforwardly determined from facts produced by the use of measurement bodies naively regarded as actually rigid. It is also a different view than Schlick's holist conventionalism, based on a stipulation of the rigidity of certain bodies that, when employed as measuring rods, lead to the “simplest physics”. Certainly, for Helmholtz. too, the axioms of geometry are propositions not regarding spatial relationships alone but also “the mechanical behavior of our most rigid bodies during motions”. But the sense in which a measurement body is considered “rigid” is that provided by notion of an ideal imaginative extension belonging to spatiality itself as a form of “outer intuition”. In the last analysis, Helmholtz's “geometric empiricism” boils down simply to his view that the propositions of pure geometry by themselves make no determinate assertions about space but only in connection with the instruments of geometrical measurement.

Then such a system of propositions is given an actual content, which can be confirmed or refuted by experience, but which for just that reason can also be obtained by experience.82

This has been rightly regarded as “a very powerful argument against the Kantian philosophy of geometry and is perhaps the main reason why the latter could not survive the discovery of non‐Euclidean geometries: a priori knowledge of physical space, devoid of physical contents, is unable to determine its metrical structure with the precision required for physical applications”.83 In a word: Helmholtz (p.74) argued against the Kantian philosophy of geometry while retaining an inherently Kantian theory of space.

Some care must be taken here, for Helmholtz also described (and then rejected) the possibility of a “transcendental geometry”, based on a “transcendental” or “inner intuition” of ideal geometrical spatial structures. The concept of such “absolutely unalterable and immobile” geometrical figures is a “transcendental concept”, formed independently of any experience of actual bodies and to which the behavior of actual bodies need not correspond.84 Relations of congruence and likeness of such figures to one another could be given in a “transcendental intuition” without the figures ever being brought into coincidence through motion, for motion “belongs only to physical bodies”.85 Now “a strict Kantian” (ein strenger Kantianer, original emphasis), Helmholtz allowed, could maintain that the axioms of geometry were therefore “propositions given a priori through transcendental intuition”. But, Helmholtz then pointed out, the axioms of such a “transcendental geometry” would no longer be synthetic, instead following analytically from the transcendental concept of the immobile fixed geometrical structures.86 To be sure, the procedure of comparison of magnitudes in “transcendental intuition” (and so, without mobility) is an idea of doubtful physical meaningfulness; our scientific and practical interest is irrevocably attached to measurable relations of spatial alikeness, not to what appears as spatially alike in “this inner intuition” (diese innere Anschauung).87 The differentiation from “a strict Kantian” is therefore instructive: Helmholtz, a nonstrict Kantian, recognized an a priori form of spatial or “outer intuition” as a condition of the possibility of geometrical measurement, but not the “transcendental axioms” governing the geometrical relations of ideal and immobile spatial figures given within an “inner” or “transcendental intuition”. Space (“outer intuition”) is transcendental without the axioms being so.

We are now in a position to see how Schlick tried to appropriate Helmholtz as an empiricist precursor for his own “empiricist interpretation of the new physics”. The talk of an a priori spatial intuition, Schlick cautioned, is to be understood only in “a non‐epistemological and psychological sense, pertaining only to the psychic makeup of the cognizing consciousness”. On the other hand, in the proper “transcendental‐logical exegesis” of Kant,

the essence of the a priori consists in its comprising the ultimate axioms which alone form the foundation for all rigorous cognition and guarantee the latter's validity.88

As he did with Cassirer, Schlick once again set the ground rules of what is, and what is not, suitable to be designated as “Kantian” (or “neo‐Kantian”). As just shown, Helmholtz had rejected the “strict Kantian's” account of a “transcendental geometry”. So, Schlick concluded, Helmholtz's “epistemology thoroughly deviates from Kant's”. To be sure, Helmholtz considered physical geometry, characterized by a set of axioms, as an empirical theory—there is no necessary geometry. But he limited the available options for empirical selection to spaces of constant curvature because of the “fact” of free mobility of fixed bodies, a restriction imposed by the a priori form of spatiality itself. In failing to give recognition to Helmholtz's attempt to retain a Kantian account of space as a necessary form of “outer intuition”, Schlick could only judge Helmholtz's discussion of “actually rigid bodies” to rest (p.75) on a definitional circularity, broken only by following the example of Einstein and making a stipulation regarding “rigid bodies”.

3.5 Conclusion

In the ensuing year, Schlick's empiricist sanitizing of Helmholtz continued apace; even the charge of “circularity” in Helmholtz's definition of rigid bodies was laid aside. Thus, in a 1921 talk, published in 1922, commemorating Helmholtz as an epistemologist, Schlick lauded Helmholtz's theory of space as true, locating the kernel of his empiricism in the recognition that

the content of geometrical principles … is thus at bottom, a physical claim: something is thereby stated about the observable behavior of bodies, light rays and so on. And if bodies had behaved differently, in certain ways, from what we actually observe, we should have adopted from the outset and assumed as correct another theory of space than that of Euclid, without being prevented from doing so by any a priori form of intuition. Helmholtz's reasons for this are chiefly founded on the indissoluable union of the spatial and the physical in experience. … His arguments are irrefutable. … His theory, widely contested at the time, has been brilliantly confirmed, of course, by the progress of science. What Helmholtz declared possible is now known to be the case: through Einstein's general theory of relativity, contemporary physics has in fact reached the conviction that natural phenomena, established by most accurate observation, compel us to attribute non‐Euclidean properties to real space.—It is a great satisfaction to the philosopher to observe that even in epistemology there is such a thing as confirmation by the advancement of science.89

The statement that “natural phenomena … compel us to attribute non‐Euclidean properties to real space” signals Schlick's new empiricism, and the end of his geometric conventionalism. To be sure, Schlick could point to Helmholtz's insistence on “the indissoluble union of the spatial and the physical in experience”, a demand also made by Poincaré. But nothing remains of the conventionalist strategy of choosing measuring bodies in such a way as to produce “the simplest physics”; rather, “geometrical principles” make “physical claims”, indeed, “about the observable behavior of bodies, light rays, and so on”. No longer can the geometrical principles be chosen independently and the physical laws adjusted accordingly. Instead, these principles are directly implicated in the “observable behavior” of the objects and processes of geometrical measurement, for the fundamental notions of metric and congruence are tied, by stipulation, to such instruments of measurement, which can therefore be considered to be independent of the basic postulates of the physical‐geometrical theory.

Schlick's attempt to situate Helmholtz as a precursor of Einstein, assimilating both to the new empiricism he needed to oppose to Cassirer's critical idealism, can only be viewed as inspired. As shown above, Einstein himself was of two minds about the procedure of tying geometry to experience through the expedient of “practically rigid bodies”. In Schlick's eyes, however, the methodology of “practical geometry” furnished a much needed weapon against the Kantian interpretations of relativity theory. His “empiricism with constitutive principles” quickly (p.76) faded from view, for its bite against the neo‐Kantian holism was toothless. In its place, indeed, came a new empiricist interpretation of physics wherein the ties of theory to observation are explicitly made through “coordinative definitions”. In chapter 4, I show that the mechanism of “coordinative definitions” was definitively stated just a few years later by Reichenbach and henceforth was associated with his name. But as will be seen there, Schlick's influence was instrumental in weaning Reichenbach away from his early neo‐Kantian theory‐specific, and so, “relative a priori constitutive principles” to a “consistent empiricism” where “constitutive principles” have become stipulations (in the case of general relativity) about rigid rods and ideal clocks. The ensuing account of the empirical determination of the metric in general relativity would become the logical empiricist paradigm of how the terms of a physical theory, regarded initially as signs within an uninterpreted logico‐mathematical calculus, received empirical content through connection to observation terms, via conventionally adopted “correspondence rules”.


(1.) HR 015-49-26; quoted with permission of the Archive for Scientific Philosophy, Hillman Library, University of Pittsburgh.

(2.) Schlick (1915), 163; Engl. trans. (1979a), 178.

(3.) Cassirer (1937), 132; (1939), 201–202.

(4.) Cassirer (1921), 71; Engl. trans. (1953), 412.

(5.) Details of Cassirer's life, see Krois (1987).

(6.) Cassirer (1906, 1907, 1920).

(7.) Cassirer (1918).

(8.) See Holzhey (1992) and Orth and Holzhey (1995) for discussion and further references.

(9.) Cassirer (1921), 2: “Albert Einstein has read the following essay in manuscript and encouraged [gefördert] it through some critical comments that he attached to the reading. I cannot let this book go out without expressing my heartfelt thanks to him again in this place”.

(10.) A. Einstein to M. Schlick (17 October 1919); as cited and translated in Howard (1984), 625.

(11.) A. Einstein to E. Cassirer (5 June 1920); EC 8-386, Einstein Archive, Jerusalem.

(12.) Schlick (1904).

(13.) Herneck (1970), 9.

(14.) Schlick (1917).

(15.) See Howard (1984) and Hentschel (1986).

(16.) Letter of Einstein to Born (9 December 1919), in Born (1969), 38.

(17.) A. Einstein to M. Schlick (17 October 1919), as cited and translated in Howard (1984), 620.

(18.) Einstein (1921a), 5, 6, 14; (2002), 387, 388, 396.

(19.) Coffa (1991), 199.

(20.) Von Laue (1921), 42.

(21.) Schlick (1921a), 98; Engl. trans. (1979a), 323.

(22.) Friedman (2000a), 115.

(23.) Schlick to Reichenbach (26 November 1920), HR 01563-22, as translated in Coffa (1991), 201–202.

(24.) Schlick (1921b), 340.

(25.) Neurath (1929), 89: “The scientific world conception knows no unconditionally valid cognition from pure reason, no synthetic judgments a priori, as lie at the foundation of the Kant's epistemology, and really all pre- and post-Kantian ontology and metaphysics”.

(26.) Schlick (1921a), 98; Engl. trans. (1979a), 324.

(27.) Coffa (1991), 204.

(28.) Carnap (1928).

(29.) Schlick (1918), 12; Engl. trans. (1985), 13.

(30.) Schlick (1918), 36; Engl. trans. (1985), 38.

(31.) Schlick (1918), 39, 42; Engl. trans. (1985), 42, 44.

(32.) Schlick (1918), 326–327; Engl. trans. (1985), 384.

(33.) Schlick (1918), 309; Engl. trans. (1985), 363.

(34.) Schlick (1918), 46; Engl. trans. (1985), 49.

(35.) Schlick (1918), 63; this text is replaced by new material in the second edition (1925), 64–69; Engl. trans. (1985), 69–75.

(36.) Schlick (1918), 44, also 46; Engl. trans. (1985), 46, also 49.

(37.) Schlick (1918), 47; Engl. trans. (1985), 50.

(38.) Schlick (1918), 58; Engl. trans. (1985), 62.

(39.) Schlick (1915), 153; Engl. trans. (1979a), 171.

(40.) Schlick (1915), 150–151; Engl. trans. (1979a), 168–169.

(41.) Schlick (1917), 183; Engl. trans. (1979a), 244–245.

(42.) Schlick (1925), 64–72; Engl. trans. (1985), 69–78.

(43.) See Howard (1994), 68–70. As Howard points out, this is the decisive step toward the modern empiricist dichotomy between analytic and synthetic statements attacked in 1950 in Quine's “Two Dogmas of Empiricism”.

(44.) Schlick (1925), 66; Engl. trans. (1985), 71.

(45.) Friedman (2002), 227, n. 63, rightly insists that Schlick distinguished between concrete definitions and conventions.

(46.) Howard (1994), 71.

(47.) Einstein (1921a), 124; cf. Engl. trans. (1983), 28. Excerpts in Feigl and Brodbeck (1953), 189–194.

(48.) Einstein (1921a), 125–126; cf. Engl. trans. (1983), 31–33.

(49.) Einstein (1921a), 126; cf. Engl. trans. (1983), 35.

(50.) Stachel (1989b).

(51.) Here, and throughout, when pertaining to extension between events in space-time, I have translated the German word Strecke as “tract”, although in the context of foundations of geometry, it normally translates as “segment”.

(52.) Einstein (1921a), 129; cf. Engl. trans. (1983), 38.

(53.) Pauli to Eddington (20 September 1923), in Pauli (1979), 118: “Certainly, the most beautiful achievement of the theory of relativity was to have brought the measurement results of measuring rods and clocks, the paths of freely-falling mass particles and those of light rays, into a determinate inner connection [Verbindung]. Logically, or epistemologically, this postulate does not admit of proof. However, I am persuaded of its correctness”.

(54.) Poincaré (1902) and Engl. trans. (1929), ch. 4. See Ben-Menahem (2001) for an illuminating discussion.

(55.) Einstein (1921a), 127; cf. Engl. trans. (1983), 36.

(56.) Einstein (1914), 1079–1080.

(57.) Weyl (1918d), 385; repr. in Weyl (1968), vol. 2, 2.

(58.) This is now a commonplace, nowhere better and more simply motivated than in the remarkable Geroch (1978).

(59.) Einstein (1923e), 483–484.

(60.) Einstein (1925a), 17 and 18–19.

(61.) Einstein (1925a), 20.

(62.) Poincaré (1902), 73; Engl. trans. (1929), 63.

(63.) Helmholtz (1921); Engl. trans. (1977).

(64.) Koenigsberger (1906), 312.

(65.) Helmholtz (1866). See Torretti (1978), 162.

(66.) Helmholtz (1868), repr. in Helmholtz (1921), 38; Engl. trans. (1977), 39.

(67.) Torretti (1978), 391, n. 8. Hilbert (1917) refers to “measure threads” (Massfaden).

(68.) Lie (1890), 466–471.

(69.) Helmholtz (1868), repr. in Helmholtz (1921), 43; Engl. trans. (1977), 44–45.

(70.) Lie (1890), 454ff. See the discussion in Torretti (1978), 161.

(71.) P. Hertz in Helmholtz (1921), 62 (n. 15 to the text of 43); Hertz observed: “One is not allowed to apply to the infinitely small the axioms which were laid down for the finite case, as S. Lie has shown”; Engl. trans. (1977), 65.

(72.) Helmholtz (1870), repr. in Helmholtz (1921), 15; Engl. trans. (1977), 15.

(73.) Schlick in Helmholtz (1921), 30; Engl. trans. (1977), 31.

(74.) Helmholtz (1870), repr. in Helmholtz (1921), 18; Engl. trans. (1977), 19.

(75.) Schlick in Helmholtz (1921), 33; Engl. trans. (1977), 34.

(76.) Helmholtz (1868), as repr. in Helmholtz (1921), 38; Engl. trans. (1977), 39.

(77.) Friedman (1997).

(78.) Friedman (1997), 33.

(79.) Helmholtz (1878), as repr. in Helmholtz (1921), 117–118; (1977), 124.

(80.) Helmholtz (1878), as repr. in Helmholtz (1921), 140; (1977), 149.

(81.) This is what I take Torretti (1978), 168, to have meant in stating, “The notion of a rigid body must therefore be regarded, if Helmholtz is right, as a concept constitutive of physical experience, that is, as a transcendental concept in the proper Kantian sense”. Unlike DiSalle (1993, 513–514; and Dingler, whom DiSalle cites), I do not see a conflict with Helmholtz's stated aim of providing a factual foundation for geometry, such facts being “constituted” and not simply (and naively, in the light of Riemann) “given”.

(82.) Helmholtz (1870), as repr. in Helmholtz (1921), 24; Engl. trans. (1977), 25.

(83.) Torretti (1978), 170.

(84.) Helmholtz (1870), as repr. in Helmholtz (1921), 23–24; Engl. trans. (1977), 24–25 and Helmholtz (1878), as repr. in Helmholtz (1921), 144; Engl. trans. (1977), 154.

(85.) Helmholtz (1878), as repr. in Helmholtz (1921), 144; Engl. trans. (1977), 154.

(86.) Helmholtz (1870), as repr. in Helmholtz (1921), 24; Engl. trans. (1977), 25; (1878) in Helmholtz (1921), Engl. trans. (1977), 157.

(87.) Helmholtz (1878), as repr. in Helmholtz (1921), 145–146; Engl. trans. (1977), 155–156.

(88.) Schlick in Helmholtz (1921), 157–158; Engl. trans. (1977), 168.

(89.) Schlick (1922b); Engl. trans. (1979a), et al.