Erich Kretschmann as a proto-logical-empiricist: Adventures and misadventures of the point-coincidence argument

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Abstract

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

Highlights

► A 1915 article by Kretschmann was the source of Einstein's point-coincidence remark. ► Einstein took from Kretschmann nothing but the expression “point coincidences”. ► Logical Empiricists unwittingly reproduced Kretschmann's and not Einstein's argument.

Introduction

In the early 1980s, Stachel (1980) and Norton (1984) famously shed new light on Einstein's celebrated, yet somewhat cryptic, claim that all physical measurements amount to a determination of space–time coincidences, such as the matching of a pointer with a scale, or, if the world consisted of nothing but particles in motion, the meetings of their world-lines.

In Einstein's published writings, this remark—which Stachel has successfully labeled the “point-coincidence argument”—amounts to the requirement of “general covariance”: since all coordinate systems necessarily agree on coincidences, that is, in everything observable, there is no reason to privilege one coordinate system over another. However, Stachel and Norton's archival work on Einstein's private correspondence unmistakably showed that he had introduced the argument in order to identify the fallacy that lurked in his now overwhelmingly famous “hole argument”: solutions of the field equations that differ only by a coordinate transformation agree on point-coincidences—that is, on everything observable—and thus they represent the same physical situation.

This “discovery” has revolutionized the historiographical work on the genesis of general relativity (cf. the volumes of the series Einstein Studies edited by Howard and Stachel and the monumental Renn, 2007, for a recent overall presentation) and engendered a vast and animated debate about its philosophical interpretation that, to this day, has shown no signs of exhaustion (cf. for instance Rickles, 2008, chap. 5, for an effective overview). Another unintended, but no less important consequence of this extensive historical work was to drastically change our perception of early Logical Empiricism. Einstein's “public” point-coincidence remark in his 1916 review paper had been previously regarded as “the beginnings of the empiricist and verificationist interpretation of science characteristic of later positivism” (Friedman, 1983, but see Friedman, 1999, p. 39; note 22). During the 1990s, a time of renewed interest in the emergence of logical-empiricist interpretation of general relativity (Friedman, 1999, Giere and Richardson, 1996, Howard, 1994), historical scholarship had an easy time showing that Einstein's point-coincidence argument, considered in the correct setting, could hardly nurture logical empiricists' eagerness “to find heroic precursors” (Howard, 1996, 121) of their verification theory of meaning.

Ryckman (1992) pointed out that the logical empiricists' “more geometrico interpretation of the point-coincidence remark” (Ryckman, 1992, 477) was “no more congenial … than a left shoe to a right foot” (Ryckman, 1992, 496). The truly philosophically revolutionary achievement of general relativity cannot be located in the fact-convention opposition—which was after all a sort of variation of the old Kantian form-content opposition—but precisely in the opposing idea that space and time have no reality independent of the matter or fields within them (Ryckman, 1992). Howard (1999) suggested that the logical empiricists had failed to “distinguish in principle unobservable, infinitesimal point coincidences from observable, finite pointer coincidences” (Howard, 1999, 493); Einstein considered “coincidences” as physically real “by virtue of their invariance properties”, not because they are directly observable (Howard, 1999, 494). Thus also the classical observable-theoretical dichotomy—another watermark of Logical Empiricism—cannot be considered a consequence of Einstein's argument.

The present paper intends to make a contribution to the history of the logical-empiricist misinterpretation of the point-coincidence argument, starting with a historical episode that, as far I can see, has been neglected among historians of the philosophy of science. In the early 1990s Howard and Norton, in a seminal paper (Howard & Norton, 1993) dedicated mainly to the correspondence between Einstein and Paul Hertz (cf. Section 2.3), suggested in passing that an article written by a then-unknown school teacher Erich Kretschmann might have been the unacknowledged source of Einstein's point-coincidence remark (cf. also Janssen, 2007).

Erich Justus Kretschmann (born in Berlin in 1887) had just gotten his doctorate under the guidance of Max Planck by attempting to provide a Lorentz-covariant theory of gravitation (Kretschmann, 1914; see Section 2.1). In December 1915 he published a two-part paper with a certain epistemological flavor (Kretschmann, 1915), in which, by relying on the work of Henri Poincaré and Ernst Mach, he argued that only “topological” relations encoded in point-coincidences are directly accessible to experience (Section 3). It was only shortly after the paper was distributed that Einstein started to use the expression “point-coincidences” in private correspondence with Paul Ehrenfest, Michele Besso and Hendrik Lorentz, in order to convince them that solutions of the field-equations that differ only by a coordinate transformation are physically equivalent (Section 2). Einstein then abruptly inserted the argument into the quite different mathematical tradition that had culminated in Ricci and Levi-Civita's absolute differential calculus (Section 4). Kretschmann himself swiftly realized this, and in August 1917 he turned the public version of the point-coincidence argument against Einstein in a paper that would make him famous (Kretschmann, 1918, 6 Kretschmann after Einstein: the triviality of the “public” point-coincidence argument).

Considered from this perspective, there is indeed a seldom noticed “ironical attachment” to the logical-empiricist reception of general relativity: in the attempt to unravel the philosophical implications of Einstein's point-coincidence remarks, the founding fathers of Logical Empiricism unwittingly ended up repeating the philosophical reflections entailed in a minor paper that an obscure Gymnasiallehrer had written before general relativity even appeared (Section 5). Kretschmann must be credited not only for having possibly suggested to Einstein the expression “point coincidences”; he also anticipated with astonishing clairvoyance the main lines of the philosophy of space and time that Moritz Schlick, Hans Reichenbach and Rudolph Carnap managed to the transform in the 1920s into the only credible philosophical approach to Einstein's new theory of gravitation (Ryckman, 2005, chap. 3). The fact/convention and observable/theoretical distinction and their entanglement with the topological/metrical opposition have a much more humble lineage than the logical empiricists were inclined to believe (Section 8).

As we will try to show, Kretschmann's “Proto-Logical-Empircism” provides a clue as to why, in the 1920s, the logical empiricists’ reading of the point-coincidence argument was so irresistibly convincing, while now it seems irremediably implausible. At the time, philosophically informed readers could not resist the sort of “Pavlovian reaction” of considering the language of “coincidences” as the heir of the tradition from which Kretschmann had taken it, namely, from the familiar work of authors such as Helmholtz, Poincaré and Mach, etc. The context of the work of Riemann, Christoffel and Ricci in which Einstein had inserted Kretschmann's parlance was far too detached from the main issues of the philosophical debate at the time to be taken into consideration.

Despite conventional wisdom, the claim that only point-coincidences are physically significant could only be properly understood by reading it out of context. When Kretschmann, inspired by Mach and Poincaré, resorted to this turn of phrase he was pointing out the “scarcity” of mathematical structure to which experience and observation have access. On the contrary, Einstein, by appropriating Kretschmann's wording, was dealing with an uncomfortable “abundance” of mathematically different solutions to the field equations that was allowed by Ricci and Levi-Civita's mathematical technique (Section 4).

To the logical empiricists, by unconsciously reading Einstein's point-coincidence argument as Kretschmann had originally presented it, general relativity indeed appeared to deprive physical reality of all but “topological” properties of the coordinate system. Yet, if any philosophical implication can be attributed to Einstein's theory, it is precisely the denial that the bare coordinate system, with its mere “topological” properties, has any independent physical reality. It has only been since the mid-fifties that Peter Bergmann—Einstein's former assistant at Princeton—restored the spirit of Einstein's notion of “coincidence”, by redefining the notion of what is “conventional” and what is truly “observable” in general relativity in a way the philosophical debate has only much later come to appreciate (Section 9).

Section snippets

Einstein before Kretschmann: Einstein's famous argument against general covariance

As is well known, in as early as 1913, after having become familiar with Ricci and Levi-Civita's “absolute differential calculus” (Levi-Civita & Ricci-Curbastro, 1900; cf. Reich, 1994), Einstein and Marcel Grossmann were able to outline a theory of gravitation based on the expression ds2=gμνdxμdxν; the coefficients gμν represented the numbers by which one has to multiply coordinate differences in order to get real distances as well as the potentials of the gravitational field at a certain

Kretschmann's early point-coincidence argument

On 15 October 1915 Kretschmann—at the time a substitute teacher in a Königsberg Gymnasium—had finished writing a rather convoluted paper in two parts, which was entitled: Über die prinzipielle Bestimmbarkeit der berechtigten Bezugssysteme beliebiger Relativitätstheorien (Kretschmann, 1915). The Annalen der Physik received Kretschmann's first post-doctoral paper on 21 October 1915. The paper contained, among others things, an argument that—unbeknownst to its inventor—would enjoy an enormous

The “Private” point-coincidence argument

Kretschmann's paper appeared in the Annalen der Physik on 21 December 1915; on 26 December Einstein wrote a letter to Ehrenfest which has been referred to in the literature many times in the last thirty years. Einstein briefly reminded Ehrefenst of the problem raised in the above-mentioned § 12 of his 1914 Sitzungsbericht — the possibility of referring two solutions G(x) and G(x) to the same coordinate system x. “In place of § 12,” he continues, one must introduce the remark that “[t]he

Between Kretschmann and Einstein: Schlick's point-coincidence argument

Thus, Einstein's formulation of the point-coincidence argument in his 1916 review article was utterly elliptical. The ambiguity resides precisely in the claim that only point-coincidences have physical reality, that is only the fact that two or more world-lines have “a particular system of co-ordinate values, x1,x2,x3,x4 in common” (Einstein, 1917, 64). As we have tried to show, Einstein borrowed Kretschmann's expression in order to explain to his correspondents that the particular coordinate

Kretschmann after Einstein: the triviality of the “public” point-coincidence argument

Just as Einstein was enthusiastic about Schlick's 1917 paper, he was also pleased with his booklet as well. In a letter to Schlick on 21 May 1917, Einstein described the “last section, ‘Relations to Philosophy’ as excellent [vortrefflich]” (CPAE 8, Doc. 343, 456). Employing the parable of two different people pursuing physics interdependently, he suggested to Schlick that he should further clarify the notion of what is real in physics: not “the immediately given”, but exclusively the

Einstein after Kretschmann once again: countering Kretschmann's objection

At the time, the correct relation between covariance and invariance was of course not so easy to grasp (Norton, 1995). On 1. February 1918 Einstein could still write to Sommerfeld that “Schlick's presentation is masterful [meisterhaft]” (CPAE Doc. 453, 627). On the very same day, the Heft of the Annalen der Physik that contained Kretschmann's paper was distributed. In a letter from 17 February 1918 (CPAE 8, Doc. 465, 649f.), Gustav Mie must have drawn Einstein's attention to Kretschmann's

Logical empiricists and Kretschmann's early point-coincidence argument

In his 1918 Allgemeine Erkenntislehre, Schlick further articulated the epistemological consequences of his “method of coincidences” (Schlick, 1918, 236) by introducing a form of “critical realism”—probably inspired by Riehl's7 work (Heidelberger, 2007). “Coincidences”—on which different senses of the same and different individuals necessarily agree—give access to nothing less than the

Conclusion

After the war, Kretschmann continued his career in East Germany as a professor of theoretical physics at the Martin-Luther University of Halle-Wittenberg (Habe, 1955, 647). Most of the European adherents of Logical Empiricism had moved to the United States in the second half of the 1930s, and had essentially abandoned their interest in relativity.

In his famous contribution (Reichenbach, 1949) to the Library of Living Philosopher's volume in Einstein's honor (Schilpp, 1949), Reichenbach still

Acknowledgments

I would like to express my gratitude to Don Howard for his permanent support and for numerous stimulating discussions. Particular thanks also go to one of the anonymous referees for valuable comments and helpful suggestions.

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