# Truth in geometry: metrical conventions and Minkowskian relativity

Scott A. Walter scott.walter [at] univ-nantes.fr, Faculty of Science and Technology, University of Nantes
In D. Flament, ed., Histoires de géométries : textes du séminaire de l’année 1996, 61–76. Paris: Éditions de la Maison des sciences de l’homme, 1998

## Introduction

The eminent French mathematician Jacques Hadamard once commented that, with hindsight, the history of analytical mechanics seemed to drive inexorably towards the general theory of relativity, and mathematicians, including himself, “all merited the same reproach”, for not having discovered this theory long before it was actually established by Einstein.11endnote: 1 Hadamard 1925: 443. One of the reasons advanced by historians to explain this putative epidemic of mathematical blindness was the hegemony of the conventionalist interpretation of the geometry of physical space.22endnote: 2 Morris Kline (1990: 921-2) admirably exemplifies this position. According to the latter doctrine, developed in the 1880s by one of Hadamard’s instructors, Henri Poincaré, it was unwise to define geometric objects in terms of physical phenomena. In this essay, I reexamine the significance of the conventionalist doctrine in the history of the theory of relativity, in support of two theses. First, leading fin-de-siècle practitioners of mathematics rejected geometric conventionalism on principle, and were consequently sympathetic to the possibility of an empirical determination of the non-Euclidean structure of space. Second, with the advent of the geometric form of the theory of relativity, and in harmony with established principles, several scientists found theoretical grounds for the rejection of the conventionalist view.

While this essay is focused on the history of the theory of relativity, it benefits from a large store of research on the history of Poincaré’s philosophy of geometry, which is a central theme in analytic philosophy.33endnote: 3 Studies I have consulted include Reichenbach; Grünbaum, Nagel; Sklar; Glymour; O’Gorman (1977); Torretti (1978, 1982); Giedymin (1982; 1991; 1992); Gillies (1986); Paty (1992); Heinzmann (1995). In the history of science, the relation of Poincaré’s philosophical beliefs to his critique of physical theories is also a subject of abiding interest.44endnote: 4 Holton, Goldberg, Paty, Darrigol. Poincaré was known by his contemporaries as one of the founders of relativity theory, along with H. A. Lorentz and Albert Einstein. What Poincaré refused, as I have argued elsewhere, was the geometrical interpretation of relativity, based on a four-dimensional metric of indefinite quadratic form, proposed in 1907 by Hermann Minkowski.55endnote: 5 Walter 1996.

## The philosophical origins of the conventionalist doctrine

The philosophy of geometry developed by Henri Poincaré has a primary source in the debates in France concerning the logical coherence and physical significance of Non-Euclidean geometry in the 1870s and 1880s. While French mathematicians were not directly involved in the reevaluation of the foundations of geometry in the late 1860s, the research of Riemann, Beltrami and Helmholtz found vocal partisans on French soil.66endnote: 6 Among most active supporters of non-Euclidean geometry in France was the Bordeaux mathematician J. Hoüel, translator of Lobatchevsky, Riemann, Beltrami and Helmholtz, and partisan of a physical interpretation of non-Euclidean geometry (see Hoüel 1875). The Carton affair, in which Joseph Bertrand was persuaded to withdraw his support of a paper in the Comptes Rendus purporting to prove the parallel postulate, marked the establishment of a the mathematical droit de cité for non-Euclidean geometry.77endnote: 7 Efforts to prove the parallel postulate did not end with that of Carton. Paul Barbarin recounted that a special Commission on Parallels was created by the Academy in order to examine the flood of demonstrations. Apparently, the Commission found no satisfactory submissions, since Carton’s proof had no successors in the Comptes Rendus.

The philosophical counterpart to this right turned out to be more difficult to secure. Thus Paul Tannery’s plaidoyer for an empirical determination of the geometry of space faced strong opposition from neo-Kantian philosophers such as Charles Renouvier and Louis Couturat.88endnote: 8 Tannery 1876. One of the arguments employed against non-Euclidean geometry was that of the “objectivity” of Euclidean geometry, corresponding to the spatial intuition which justified the latter’s status as an ideal science and exemplar of synthetic a priori knowledge.99endnote: 9 For a recent discussion of neo-Kantian responses to non-Euclidean geometry, see Panza 1995.

In 1891, a radical response to the neo-Kantian argument came from the new star of French mathematics, Henri Poincaré, who proposed a termwise translation of basic objects of Euclidean geometry into the language of non-Euclidean geometry (Poincaré 1891).1010endnote: 10 Some of Poincaré’s ideas on the foundations of geometry may be found in his earliest known manuscripts on automorphic functions, see Gray and Walter, eds., 1997). The question of which geometry was “true”, based on its empirical adequacy, was thereby rendered moot.1111endnote: 11 Poincaré refined his ideas on the conventionality of geometry in the 1890’s, extending them to theoretical constructs in general. He also constructed arguments to support the tridimensional nature of space, while recognizing that this was dependent on the choice of basic elements (points instead of spheres, cf. Lie). His writings were collected and published in 1902 under the title La Science et l’Hypothèse. This volume went through several editions, and was translated into German and English by 1905, when a second volume of philosophical writings appeared, followed by others. A knock-down response to the argument of the apparent absence of any physical manifestation of non-Euclidean geometry, Poincaré’s astute technical trick was much admired by philosophers, both in his day and after.

From the start, Poincaré’s philosophy of geometry was indebted to Helmholtz, in that Poincaré considered the origin of geometry to be found in the notion of the motion of solids. Like Helmholtz, Poincaré originally held that solid objects existed in actual space. These solids allowed an approximate determination of the group of transformations corresponding to the geometry of physical space (Poincaré 1887:91).1212endnote: 12 “…il existe dans la nature des corps remarquables qu’on appelle les solides et l’expérience nous apprend que les divers mouvements possibles de ces corps sont liés à fort peu près par les mêmes relations que les diverses opérations du groupe choisi.” Poincaré 1887:91. For reasons unknown, sometime between 1887 and 1891 Poincaré decided that solid objects did not exist in physical space. This decision may have led him to the more extreme position concerning the geometry of space, expressed in Poincaré 1891. The fact that approximations were inherent to the act of measurement led Poincaré to conclude that the geometry of space was undecidable in any exact sense (Poincaré 1891). The very question of the geometrical structure of space was nonsensical, or “unmeaning” (Poincaré 1906). In a series of philosophical essays, Poincaré developed and refined his position in response to criticism by professional logicians such as Bertrand Russell, introducing a variety of new arguments, notably from the new field of Analysis situs (Poincaré 1895).

There are two aspects of Poincaré’s writings that I wish to underline. First of all, Poincaré was of the opinion that Euclidean geometry would remain the most convenient geometry for physics, in virtue of the simple form the expression of the laws of Newtonian mechanics took on, in this case only. In the second place, Poincaré claimed that geometry was not an experimental science. Here, he parted company with Helmholtz, in that for Helmholtz, the geometry could be determined by experiment on the condition that to the axioms of geometry some part of mechanics be adjoined.1313endnote: 13 Helmholtz 1876. For Poincaré, such a move was pointless, because the objects of mechanics corresponded only approximately to physical phenomena.1414endnote: 14 Helmholtz was aware of the approximations inherent to measurement, see Helmholtz 1867: 372.

Poincaré’s philosophy appealed to many philosophers (Paul Natorp, Aloys Müller), and a few philosopher-physicists (Philipp Frank, Ernst Cassirer), yet mathematicians were not willing in general to subscribe without reserve to Poincaré’s philosophy. Few, if any, denied the possibility of creating a termwise dictionary to translate elements of Euclidean geometry into hyperbolic, or elliptic geometry, yet to my knowledge, no other leading mathematician was willing to deny that space had a geometry.

In the next two sections, some applications of non-Euclidean geometry in nineteenth century physics and mathematics are discussed, with an emphasis on their situation in a socio-cultural context. For brevity, I will refer to the position according to which the geometry of physical space had no meaning, as Poincaré’s doctrine.

### Fin de siècle physicists and non-Euclidean geometry

The need in physics for a non-Euclidean geometry was by no means evident circa 1905. The abstraction of non-Euclidean geometry was itself an obstacle for some physicists. Clerk Maxwell’s reaction to Riemann’s Habilitationsvortrag (translated by William Kingdon Clifford in 1873), for example, was to question the pertinence of his definition of coordinates (Harman 1982: 97). A similar attitude was displayed by J.B. Stallo in his Concepts and Theories of Modern Physics (1890).

Beyond the technical abstraction, in the empirical domain, Karl Schwarzschild’s measurement of star parallax in 1900 confirmed the conclusion drawn earlier by Lobachevsky: If space was indeed either hyperbolic or elliptic in structure, its radius of curvature was very large. The limits to goniometric precision were such that the hypothesis of the Euclidean structure of space appeared justified, and was certainly not refuted on astronomical grounds (Barbarin 1928); Jammer 1960: 163). Speculation on the curvature of space, especially since it could never be ruled out on empirical grounds, was difficult to silence.

Apart from astronomical observations, in late nineteenth-century physics the possibility of a physical manifestation of non-Euclidean geometry was encountered on occasion in relation to speculations on the dimensionality of space.1515endnote: 15 Precursors to Minkowski have been investigated in Bork (1964). Often these considerations proceeded on the (erroneous) assumption that a physical manifestation of non-Euclidean space presupposed that this was embedded in an Euclidean space of four (or more) dimensions. Thus, the Leipzig physicist Friedrich Zöllner speculated on the connection between Riemannian space of n-dimensions and the behavior of electrical atoms in a Weberian scheme of electrodynamics. As early as 1875, W. K. Clifford imagined a reduction of physics to a geometry of matter in a space of variable curvature, where local distortions were propagated “after the manner of a wave” (Clifford 1875). Also in England, Karl Pearson advanced the hypothesis of ether squirts to explain gravitation. Other theorists invoked a space of dimensionality greater than three to account for various chemical phenomena.

Speculations along these lines were sanctioned by Ernst Mach (Erkenntnis und Irrtum), who felt physicists were not compelled to locate non-sensuous objects such as atoms and molecules in ordinary space. The vast majority of these conjectures never went beyond paper, although it appears that a number of laboratory investigations by German chemists were motivated, at least in part, by the concept of higher-dimensional space.

In physics, however, such experimental investigations were unknown, and the theoretical reformulations in 3 + n-dimensional space remained extremely rare. The possibility of such a reformulation was manifest in Heinrich Hertz’s Principles of Mechanics. Yet Hertz–a paradigmatic figure of theoretical physics, in virtue of his combination of qualitative experimental investigations with elegant mathematics–employed n-dimensional space as an analytical device (after having postulated the existence of a Galilean space and absolute time). A number of literary variations on the theme of 3 + n-dimensional space and non-Euclidean geometry appeared around the turn of the century, many of which were inspired by the writings of the mathematician Charles Hinton (including, plausibly, the stories by H. G. Wells).1616endnote: 16 Boucher 1903; Jouffret 1903, 1906. While Hinton’s writings and ideas circulated widely, their influence on the scientific community is difficult to judge. Yet popular and scientific enthusiasm for the fourth dimension attained proportions significant enough to attract the attention of Henri Poincaré, who commented that an effort to translate the laws of physics in the language of four-dimensional geometry, “ce serait se donner beaucoup de mal pour peu de profit” (Poincaré 1907: 15).

Although the pages of physics journals contained few examples of such an effort prior to 1908, a warning similar to Poincaré’s was issued by Max Abraham and August Föppl in their widely-studied manual of electricity; these authors viewed physical theories in non-Euclidean and higher-dimensional spaces as unlikely (Abraham & Föppl 1905: 435). However, this opinion conflicted with one advanced earlier by Föppl, and did not succeed in putting a halt to such speculations.1717endnote: 17 I thank Olivier Darrigol for pointing out Föppl’s earlier speculation on the existence of four-dimensional space, in the first edition of his treatise on Maxwell’s theory of electrodynamics (Föppl 1894).

In summary, throughout the period leading up to the discovery of the general theory of relativity (1915), most German physicists were undoubtedly, as Arnold Sommerfeld put it, “blissfully unaware” of non-Euclidean geometry (Sommerfeld 1919). That is not to say that they were ignorant of its existence, of course, or of its ramifications for neo-Kantian philosophy; physicists simply ignored mathematical techniques which had no clear relevance to their activity. Students of theoretical physicists, including Einstein, were typically exposed to at least to the rudiments of non-Euclidean geometry, in the form of Gauss’s geometry of curved surfaces. Those with PhD’s in mathematics such as Sommerfeld and Gustav Mie may also be counted as exceptions in this respect.

### Fin de siècle mathematicians and physical geometry

The limited familiarity of physicists with non-Euclidean geometry contrasts sharply with the case for their mathematical colleagues. While the consistency of non-Euclidean geometry worried some mathematicians–the Cambridge mathematician Arthur Cayley, for example–even after the discovery of Euclidean models, both the correspondence established with ordinary geometry by the latter, and their appeal to visualization gave further impetus to the diffusion of the idea of “curved space”. Physical space, or the “space of experience”, served also as an important–interpreted–model, yet, as mentioned above, observations of stellar parallax revealed no significant large-scale curvature. Even so, the very possibility of such measurements lent plausibility to the idea that the actual geometric structure of space could be determined empirically. Indeed, from the time of its discovery, non-Euclidean geometry was presented as the “science of space.”1818endnote: 18 Cf. Johann Bolyai, “The Science Absolute of Space,” ca. 1833, translated by Halsted, reprinted by Dover.

The influence of the physiologist and physicist Hermann von Helmholtz’s writings on the foundations of geometry convinced many of the pertinence of Riemann’s differential-geometric approach. In a limpid style, Helmholtz showed the necessity of Riemann’s expression for distance in a space of constant curvature, given the possibility of motions of solids (Helmholtz 1876). The determination of the geometry of physical space, Helmholtz realized–after learning of Beltrami’s model of non-Euclidean geometry–required that some part of mechanics had to be adjoined to geometry, for example, the law of inertia. Once this was done, Helmholtz claimed, one could empirically determine that the geometry of actual space was Euclidean.1919endnote: 19 For a discussion of Helmholtz’s work on the foundations of geometry, see Richards (1977).

Riemann’s lecture on the foundations of geometry became better known, thanks in large part to Helmholtz’s arguments. Just as Riemann’s analysis of the concept of space was criticized for paucity of philosophical sophistication, Helmholtz’s papers were found wanting in mathematical rigor by professionals such as Klein, and Sophus Lie, even if Lie later borrowed upon Helmholtz’s notion of the possibility of motions of a solid body, to form the basis of his own work on the foundations of geometry.2020endnote: 20 On the philosophical reception of Riemann’s lecture, see Nowak 1989. On Klein’s and Lie’s critiques of Helmholtz, see Richards 1977: 252.

Among mathematicians, the published version of Riemann’s Habilitationsvortrag (1867) remained the work of reference for foundational questions in physical geometry. Along with the speculations by Zöllner and Clifford, mentioned above, Riemann’s work found an immediate, if limited, audience, inciting work on non-Euclidean statics and mechanics by de Tilly, Schering, and Killing, among others.2121endnote: 21 Bonola 1912; Ziegler 1985; Lützen 1993. The relation of differential geometry and analytical mechanics in the late 19th century has been the object of studies by Lützen (1995), and Tazzioli (1993).

In the hands of Riemann and Helmholtz non-Euclidean geometry had profound consequences for the foundations of geometry, but its broader significance for the practice of mathematics had yet to be demonstrated. The required apodictic application arrived in the domain of linear differential equations. Young Henri Poincaré, then in his first year as a lecturer in mathematics on the science faculty in Caen, discovered the theory of what he called Fuchsian functions, the establishment of which called upon conformal transformations in the hyperbolic plane. Poincaré’s powerful theory allowed for a parametric representation (by Fuchsian functions) of any algebraic curve, and the integration (by Zétafuchsian functions) of every linear differential equation with algebraic coefficients (see Gray 1982 and 1986; Dieudonné 1982).

The novelty of Poincaré’s method was such that mathematicians of the stature of Hermite and Klein were at first unable to grasp the relation between these functions and hyperbolic geometry, but the publication of a series of articles in the Comptes Rendus and Acta Mathematica rendered the reasoning more accessible.2222endnote: 22 Transcriptions of letters to Poincaré from Hermite and Klein appear in the Cahiers du Seminaire d’Histoire des Mathématiques. In the period following the publication of Poincaré’s work, non-Euclidean geometry came to be seen in a different light altogether: no longer a logical curiosity, non-Euclidean geometry was henceforth a standard tool in the mathematician’s analytical repertory (Houzel 1992).

Linked to the latter development, by the early 1890s the number of mathematicians interested in non-Euclidean geometry had grown to the point where this domain could be considered a sub-discipline of mathematics. Previously, non-Euclidean geometry may have been evoked in a course on projective geometry, for example, or the theory of invariants; now, non-Euclidean geometry was introduced to the university curriculum as a self-contained topic.2323endnote: 23 Course listings in the JDM-V indicate that in the first years of the 20th century, lectures on non-Euclidean geometry could be heard at Griefswald, Königsberg, Leipzig, Marburg and Münster. In England, Alfred North Whitehead taught this subject intermittently at Cambridge, starting in 1893 (June Barrow-Green, private communication), while in Paris no regular courses were given on non-Euclidean geometry per se (cf. the listings in L’Enseignement mathématique). During this period, special aspects of non-Euclidean geometry were considered as appropriate subjects for doctoral dissertations. Among the first mathematicians to lecture on non-Euclidean geometry, Felix Klein spoke of this as a “discipline of the real”–not to be confused with the “abstract mathematical views” to which it had given rise.2424endnote: 24 1890: 72. Klein lectured on non-Euclidean geometry in 1889, see Klein 1892.

Given the magnitude of Poincaré’s role in the historical development of non-Euclidean geometry, his subsequent circumscription of the domain of application of this same geometry, as mentioned above, presents a certain historical irony. If Poincaré’s doctrine had roots in French philosophy of mathematics, as I suggest, this did not imply, of course, that other French mathematicians thought like Poincaré did. In fact, a number of these mathematicians–especially those with an interest in physics–disagreed with Poincaré’s doctrine.2525endnote: 25 Jacques Hadamard and Émile Picard were perhaps the most notable of these. Among the less eminent French mathematicians, opinion was divided. Lycée professor Paul Barbarin and Hermann Laurent (examiner at the Ecole Polytechnique) spoke out against Poincaré’s doctrine, yet Poincaré found ardent defenders in the Lycée professors J. Richard, and Adolphe Buhl (1911). . Those whose interests tended more towards geometry than to physics also refused to see non-Euclidean geometry relegated to the domain of abstraction.2626endnote: 26 Barbarin (1902: 69ff); Laurent (1906: 40).

In Germany too, many mathematicians disagreed with Poincaré’s philosophy of geometry. Felix Klein’s position, while perhaps not representative of opposition to conventionalism, offers some perspective. Klein’s view was compared to that of Poincaré by Federigo Enriques in his Encyklopädie article on the principles of geometry: both mathematicians recognized a gap between the content of the geometric postulates, on one hand, and experiment and intuition on the other. Yet Enriques singled out Poincaré’s nominalism for criticism, and concluded that Riemann and Helmholtz were correct in considering geometry as a branch of physics (Enriques 1911: 6). The latter view was closer to that advanced by Klein, when he wrote that geometry could be understood as a physical science, and that, furthermore, this entailed the introduction of a class of approximation techniques–more applied mathematics, in other words (Klein 1902).2727endnote: 27 Enriques’s criticism of Poincaré’s conventionalist position on geometry was adopted by Eduard Study in his appeal for realism (1914).

This is not to say that the geometry of space occupied the center of mathematicians’ interest–on the contrary, it was a marginal area of research, even within the domain of non-Euclidean and n-dimensional geometries.2828endnote: 28 Sommerville’s bibliography of non-Euclidean geometry (1911), the most complete of its kind, is one index of the relative interest of different fields and sub-fields in this domain, indicating also their chronological development. Other aspects of non-Euclidean geometry were of greater interest to mathematicians, including the axiomatic foundations of geometry, which were developed in Italy with Pasch, Peano and Pieri, and elsewhere Veblen, Hilbert, and H. Weber. And at least until 1909, it was still possible to write a general treatise on non-Euclidean geometry excluding all discussion of physical applications.2929endnote: 29 See, for example, Coolidge (1909).

The axiomatic foundations of Euclidean geometry found a particularly cogent expression in David Hilbert’s Grundlagen der Geometrie (Hilbert 1899). Transcriptions of many of Hilbert’s lectures on this subject have been preserved, offering insight to the development of Hilbert’s ideas.3030endnote: 30 For a study of the Hilbert’s views on geometry which makes excellent use of this source, see Majer 1995. When he was a Privatdozent in Königsberg, Hilbert taught a course on projective geometry, in which he discussed the relation between physics and geometry:

The results of these domains (number theory, algebra, function theory) can be achieved by pure thinking…. Geometry, however, is completely different. I can never fathom the properties of space by mere thinking, just as little as I can recognize the basic laws of mechanics, the law of gravitation, or any other physical law in this way. Space is not a product of my thinking, but is rather given to me through my senses. Therefore I require my senses for the establishment of its properties. I require intuition and experiment, just as with the establishment of physical laws (Hilbert 1891: 6-7).3131endnote: 31 Hilbert 1891: 6-7, in Majer’s translation (1995: 263).

Such a view of the properties of space, and on the role of intuition in geometric thought held much in common with that expressed by Klein a few years later. And from 1895, Hilbert joined Klein on the Göttingen faculty; together, they established this provincial university as the acknowledged center of research and education in mathematics. From 1902, Klein and Hilbert were joined by a third full professor of mathematics: Hermann Minkowski.

#### Institutional changes in mathematics and physics

So far I have mentioned only philosophical aspects of mathematical opinion on the foundations of geometry. Yet mathematical criticism of Poincaré’s doctrine may be linked to a number of institutional developments in German science.

Perhaps the most immediate objection to a conventionalist view of the foundations of geometry was that it formed an obstacle to both theoretical and empirical investigations of the nature of physical space; a subject which had established a certain degree of autonomy by the end of the nineteenth century. The extent to which scientists modified their field of investigation as a consequence of Poincaré’s doctrine is not known, yet there are traces of such an influence. An example may be found in one of the first general textbooks on non-Euclidean geometry to be published in Germany, written by Wilhelm Killing’s young colleague in Leipzig, Heinrich Liebmann. Placing the geometry of physical space (Erfahrungsraum) in the final chapter of his treatise, Liebmann explained that he had included these considerations “in spite of the skepticism of Poincaré’s remarks” (Liebmann 1905: v).

German mathematical opposition to Poincaré’s doctrine was most deeply rooted in Göttingen, where Gauss, Weber, Riemann and Dirichlet had established a strong tradition linking mathematics and physics.3232endnote: 32 On the emergence of theoretical physics in Göttingen see Stichweh 1984, and Jungnickel & McCormmach 1986. Klein’s relation to the Göttingen tradition is described at length by David Rowe (1989); for later developments see Mehra & Rechenberg 1982: 262. In 1885, the former student of Julius Plücker and Alfred Clebsch, Felix Klein accepted a call from Göttingen, where he soon sought to assume and direct this tradition.

In a tribute to Riemann’s influence on modern mathematics, Klein soft-pedaled the criticism of geometric intuition implicit in Riemann’s lecture on the foundations of geometry. In spite of the “perfect accord” between Riemann’s developments of trigonometric series, and Weierstrass’s methods, Klein was “unable to imagine” that Riemann ever considered geometric intuition as contrary to the mathematical spirit. By pointing out Riemann’s sustained interest in problems of theoretical physics, Klein was able to argue that, in general, the mathematical results which endured were those discovered in virtue of some “concrete” representation (Klein 1890). This was the main point of Klein’s lecture on Riemann’s significance for modern mathematics, which established the continuity of his own vision of the future of mathematics with that of Gauss, Riemann and Dirichlet. Indeed, Arnold Sommerfeld, who earlier in his career had been Klein’s assistant, and then lecturer in mathematics at Göttingen, later described this university’s mathematical heritage as the “Riemann-Dirichlet-Klein tradition” (Sommerfeld 1943: vii).

Near the end of the century, Felix Klein considered the heroic period of science to be over; henceforth, Klein said, progress was to be obtained through collaborative efforts.3333endnote: 33 Klein’s own productive period in mathematical research had also reached its end, as he no doubt recognized. Typical of the sort of project Klein had in mind was the Encyklopädie der mathematischen Wissenschaften, a vast enterprise spanning decades and careers, in which leading mathematicians and physicists summarized the historical evolution and current state of the art in the multiple branches of analysis, arithmetic, algebra, geometry, mechanics, physics, and astronomy.3434endnote: 34 On the origins of this project, see Dyck (1898). Assigning himself the editorial responsibilities for the volume on mechanics, Klein confided the crucial task of editing the physics volume to his protégé, Arnold Sommerfeld, then professor of mathematics at the Bergakademie in Clausthal.

In the first years of the twentieth century then, the Göttingen tradition in mathematics and physics was flourishing anew. Felix Klein’s campaign for applied mathematics was carried out by his scientific “general staff”–Klein’s former students were placed in positions of direction in the new technological institutes, created with the wealth of German industrialists eager to reap a profit from the developments these institutes promised to provide.3535endnote: 35 On the creation of the Göttingen institutes of applied mathematics see Rowe 1989. The strategic value of these institutes for advanced research in military technology was, perhaps, a consideration as important as the economic one, if not more so in the Wilhelmian era; Ludwig Prandtl’s aeronautics institute, for example, contributed to the advance of German aviation.

David Rowe has portrayed Klein’s action in developing institutes of applied mathematics as a response to the increasing hegemony of physicists.3636endnote: 36 Rowe 1989. With the construction of new physical institutes in Germany during the period 1870-1914, the physics discipline underwent what David Cahan has called an institutional revolution.3737endnote: 37 Cahan 1986. Partly as a result of the increase in the institutional demand for a single individual capable of lecturing both on experimental physics and mathematical physics, a new physical sub-discipline was spawned: theoretical physics.3838endnote: 38 Jungnickel and McCormmach 1986. While the discipline of mathematics also benefited from the factors underlying this expansion (demographic ones, in particular), the profound institutional changes in physics may well have been perceived by Klein and others as a menace to the future health of mathematics.3939endnote: 39 David Rowe (1989:187) suggests that both Klein and Hilbert perceived a threat to their respective visions of the future of mathematics. Klein’s turn towards the development of applied mathematics may be understood, at least in part, as a response to the institutional growth of physics.

Another important development in mathematics influenced Klein’s management of exact science in Göttingen. Klein expressed concern over the over the “arithmetization” of mathematics, and in general the decline of geometry among the younger generation of German mathematicians. In order to counter this tendency, and restore the role of intuition in geometry, he sought, without success, to engage the services of the Italian geometer Gino Fano (Gray 1994: 151). Of course, Klein’s concern was not shared by all–least of all by those responsible for this development of “pure” mathematics at the expense of a mathematics attentive to applications in the natural sciences–the Berlin school of Weierstrass and Kronecker (Rowe 1986: 432).

The advent of pure mathematics influenced the scientific status of mathematics: was mathematics now to be considered a science (Naturwissenschaft) or one of the “moral sciences” (Geisteswissenschaften)?4040endnote: 40 For the origins of the term Geisteswissenschaft, see Koehnke (1986). Mathematicians were divided in their responses; many preferred, no doubt, to occupy the middle ground, like the biographer of Helmholtz and friend of Max Planck, Leo Koenigsberger, for whom mathematics was both a natural science and a moral science (Koenigsberger 1913).4141endnote: 41 A similar sentiment was expressed in France by Émile Picard (1911).

An altogether different response to the question of the place of mathematics in the grand scheme of science was proposed by the director of the Kiel observatory, the Geheimrat Paul Harzer. Invited to contribute his lecture “Stars and Space” for publication in the organ of the German Mathematical Society, of which Geheimrat Klein was then president, Harzer reviewed the question of the geometrical structure of space, recalling Schwarzschild’s efforts to quantify the curvature of space by measuring stellar parallax. From the advent of non-Euclidean geometry, astronomy rendered geometry a physical science, because

[n]o special value of the curvature can appear [..] as correct a priori; only experience can show which curvature value is valid in the geometry of actually-existing space. Thereby is geometry snatched away from the preferential position of an a priori science, and its place indicated as a science of experience among the other sciences of this kind, analytical mechanics and the exact natural sciences, among which it may indeed claim a prominent position as an ever-ready, powerful complementary science (Harzer 1908).

The empirical determination of the structure of space, according to Harzer, likewise determined the scientific status of geometry, simultaneously demoting it from the ranks of purely analytic science, and promoting it as a powerful and ready resource to the other exact sciences.

One more institutional factor related to mathematicians’ views on the relation between mathematics and physics remains to be discussed: enrollment figures in university mathematics. In Germany, the rate of increase in the number of students choosing to study higher mathematics surpassed the need for trained mathematicians. It was estimated that there were 250 jobs available annually for the 400 graduates in higher mathematics. In presenting the enrollment figures to the German Society of Mathematicians, Arthur Schönflies regretted that there was no way of restraining students’ attraction to mathematics.4242endnote: 42 In 1906, Prussian universities had an enrollment of 1600 students in mathematics, a figure which climbed at a rate of six percent or more until at least 1911, by which time the entry of female students had pushed enrollment to over 2000. See Schönflies 1911: 27-8: PZ12: 311. Student enthusiasm for mathematics in Germany was further stimulated by the development of the theory of relativity (Max Abraham, 1912). A comparison with the enrollment trends in England may be instructive here, as Cambridge mathematicians were faced with declining student interest in mathematics, and increasing interest in the study of physics; see D.B. Wilson, HSPS. See also Warwick’s study of the development of numerical mathematics in England, and the establishment ca. 1912 of a laboratory of experimental mechanics in Edinburgh (1995). In France, attention was focused at this time on the decreasing number of foreign students in French universities, as more and more of these students preferred the German university system; see Hadamard, ca. 1905. The fact that a laboratory of experimental mechanics was also established at the Sorbonne by Gabriel Koenigs, ca. 1912, may suggest there was a renewed interest in applied mathematics in Paris at this time. Mathematics classes, one read in the research journals, were overflowing with students.4343endnote: 43 Anon., “Starke Überfüllung des höheren Lehrfaches mit Kandidaten der Mathematik,” Physikalische Zeitschrift 12 (1911): 311.

What does this have to do with physical geometry? Some German mathematicians, like Schönflies, would have liked to restrain the enthusiasm of students for mathematics, others, like Klein and Runge, stimulated the demand for mathematically-trained graduates by developing institutes of applied mathematics. The physical relevance of geometrical methods, I want to suggest, was necessary, in some sense, to Klein’s program for applied mathematics, as mentioned earlier.

Additional employment opportunities for mathematicians, or at least the potential for such employment, revealed themselves in the wake of Hermann Minkowski’s geometrization of the theory of relativity. At the height of mathematical interest in Minkowski’s theory in 1911, Runge sent out a questionnaire on behalf of the International Commission on the Teaching of Mathematics (presided by Felix Klein) to his colleagues in mathematics and physics around the world, with the object of determining the place of mathematics in the university education of physicists. The Commission sought out information and opinions on several aspects of a physicist’s mathematical education, including the desirability of its extension (or reduction). The authors of the questionnaire were curious to know, in particular, if mathematicians taught courses in mechanics and “modern subjects of mathematical physics.” Reporting the results to the Fifth International Mathematicians’ Congress in Cambridge, Runge noted the “general opinion” that this area was in need of reform, and suggested that the road to improvement lie in introducing students to the “more modern views of Physics on Electricity and Matter”. Here, Runge was probably referring to electron physics, whose foundations were laid out by Joseph Larmor and H. A. Lorentz, and developed primarily by Runge’s colleagues Emil Wiechert, Max Abraham, Karl Schwarzschild, Gustav Herglotz, Arnold Sommerfeld, Walter Ritz and Hermann Minkowski in the preceding decade. In his closing remarks, Runge warned of the lurking menace to mathematics in the absence of reform (p. 602):

The main danger is the gap between physicists and pure mathematicians that seems to be widening. Mathematics is suffering from over-specialization and is cutting itself off from Natural Philosophy and Experimental Science. [..] What is really wanted is that Mathematical teachers should understand the problems and the needs of the Physicist.

The nature of the questions posed by Klein’s commission suggests that the these mathematicians saw an opportunity for members of their discipline to capitalize on the recent mathematization of relativistic physics, in extending their control over lectures offered to physics students–an impression further supported by Runge’s conclusion.4444endnote: 44 For a description of Minkowski’s contribution to the theory of relativity as an extension of the disciplinary frontier of mathematics, see Walter (1996).

The institutional revolution in physics, concern over a decline of German research in geometry and the estrangement of mathematics from physics, the evolving status of mathematics in the scientific hierarchy, and the perceived surfeit of graduates in mathematics all bear upon the debates concerning the foundations of geometry. The historical link between mathematics and physics had been sundered–Poincaré’s doctrine made this evident. The severance of this link, the French politician, academician and amateur philosopher of science Charles de Freycinet warned, in his widely-reviewed book De l’Expérience en Géométrie, would turn geometry into a “sterile science” (Freycinet 1903:128-9).

Active at the forefront of research in mathematics and theoretical physics, Poincaré was often led to consider the relations between these sciences. As a result of his provocative remarks on the ideal nature of geometry, Poincaré was understood to have established the line demarcating “pure mathematics” from mechanics and theoretical physics.4545endnote: 45 Tannery 1903: 392. The widespread rejection of Poincaré’s conventionalist stance on physical geometry, I suggest, was not merely a question of metaphysics: it was also understood as a means of ensuring the prosperity of the field of mathematics. In other words, there was a broad understanding, both in Göttingen and elsewhere, that the future of mathematics depended upon the relevance of geometry to physics. 4646endnote: 46 The rejection of Poincaré’s doctrine was compatible with a more formal, abstract approach to mathematics, as may be seen from Hilbert’s example.

## Physical geometry and the theory of relativity

Into this context, briefly sketched, of intellectual and institutional changes in physics and mathematics, appeared the first publications outlining the theory of relativity. Both Einstein’s and Poincaré’s early papers remarked the metrical change involved in defining length by light signals, and the group nature of the Lorentz transformation, but further geometric consequences were not drawn by either author.4747endnote: 47 Comparisons of Einstein and Poincaré may be found in Miller 1981 and Paty 1992.

Hermann Minkowski took up the theory of relativity at this point. One of the consequences of Minkowski’s recasting of the theory of relativity was the interest it stimulated among physicists and mathematicians, both in Germany and elsewhere. When Minkowski presented his four-dimensional space-time theory of relativity in the famous 1908 lecture in Cologne entitled “Space and Time”, Einstein’s theory was little known, poorly understood, and experimentally disconfirmed. In the aftermath of the Cologne meeting of the German Association of Natural Scientists and Physicians, the fortunes of relativity theory took a decisive turn, on the strength of two new elements: Minkowski’s geometric interpretation of the Lorentz transformation, and the news of electron-deflection measurements corroborating anew the expectations of the theory.

Minkowski’s theory was not without its detractors in physics, of course, particularly in its first two years. Early skeptics of Minkowskian relativity included Albert Einstein and Jakob Laub, who saw no advantage to be gained in a four-dimensional formulation of physics. In addition, they were convinced, like Max Abraham, that Minkowski’s definition of the ponderomotive force density was incorrect. Others, like Max Planck and Wilhelm Wien, who admired the formalism of Minkowski’s theory, reserved final judgment for the eventual experimental confirmation of the theory.4848endnote: 48 Planck 1908; Wien 1909.

In the what follows I will discuss some of the ways in which questions of physical geometry came to be viewed in the wake of Einstein’s and Minkowski’s writings on relativity.

### The conventionality of geometry in the reception of Minkowskian relativity

As mentioned above, at the time of the discovery of the theory of relativity, many mathematicians disagreed with Poincaré’s doctrine, and continued to believe that the question of the geometry of physical space was decidable in principle if not in practice. In what follows I examine the ways by which Minkowski’s theory was related to physical geometry in the years leading up to the Great War.

Prior to Minkowski’s contributions, Einstein had already modified the terms of the debate on physical grounds. As a consequence of the principles of relativity and energy conservation, Einstein deduced the equivalence of mass and energy, now symbolized in the well-known formula $E=mc^{2}$, and eventually drew the conclusion that a light ray would be bent in a gravitational field (1907: 461).4949endnote: 49 The notation has been changed; Einstein first noted the mass-energy equivalence in 1905, and expressed this in 1907 with the formula $mu=E_{0}/V^{2}$ (1905:641; 1907a: 384). On Einstein’s later derivations of the gravitational deflection of light, see Earman and Glymour 1980: 55ff. The hypothesis was not exactly new–Newton conjectured in the Opticks that light was subject to the effects of gravitation. Closer to Einstein, Planck suggested in June, 1907, that if “latent energy” should gravitate, then inertial mass would be only slightly different from ponderable mass (CP2: 207)). The notion that a beam of light was massive intrigued a few physicists, yet few astronomers demonstrated any interest, a fact bemoaned by Einstein.5050endnote: 50 Gilbert Newton Lewis and Richard Tolman (1909:713); Einstein to Freudlich ca. 1911.

This particular consequence of the special theory was not widely diffused, and its influence on discussions of the foundations of geometry appears quite limited. Only after the increase in interest in the theories of gravitation around 1912, and particularly in the aftermath of the Einstein-Grossmann theory (1913), did a few mathematicians begin to take note of the new relation between light, space, and gravitation.5151endnote: 51 Federigo Enriques, for example, described Poincaré’s position on physical geometry as “overextended”, since there were physical hypotheses of a non-arbitrary nature (such as an influence of matter on the propagation of light) capable of verification, leading to the conclusion that the geometry of space was non-Euclidean (Enriques 1913: 36, 41). Indeed, with the general theory of relativity (1915), Poincaré’s view came to be understood as completely discredited in favor of that of Helmholtz. See Einstein 1921, 1926.

The emergence of the theory of relativity coincided with the publication of the German and English translations of Poincaré’s first collection of philosophical essays, La Science et l’Hypothèse, which was the occasion for commentators to extemporize on Poincaré’s doctrine. Paul Mansion, the editor of the Belgian review Mathesis, was more direct than most in his criticism. To uphold the conventionality of geometry, Mansion said, was to deny the possibility of performing a measure of distance. This amounted to “denying all possibility of a quantitative knowledge of nature,” but Mansion doubted that “anyone would want to go so far” (Mansion 1905: 5).

Over the years, as Mansion’s remark suggests, the amplitude of the opposition, both philosophical and mathematical, to Poincaré’s doctrine had taken a personal turn. Undoubtedly, Poincaré’s patience had been worn thin. Returning once more to the question of physical geometry in the pages of the Revue de Métaphysique et de Morale, Poincaré presented the choice to be made in the following way: one could either identify the straight line with a physical process, such as the path of light, or refuse this identification. To make the first choice, Poincaré remarked, would be stupid, and that for two reasons. In the first place, if straight lines were to be defined by “curved” light paths, this would contradict another physical definition of the straight line, determined by the axis of rotation of a solid object.5252endnote: 52 Poincaré did not entertain the possibility that the axis of rotation of a non-Euclidean solid coincide with a non-Euclidean geodesic described by the path of light. Secondly, he pointed out that there was no guarantee that the physical process associated with the light line remained invariant with respect to time (Poincaré 1906a).5353endnote: 53 Poincaré’s position remained consistent on this point, at least since he reversed his initial belief (Poincaré 1887) in the physical reality of solid objects in 1891. In sum, an optical geometry was problematical, which meant that there still existed no satisfactory empirical definition of the straight line.

Now, in his original papers developing Lorentz’s theory of electrons, Poincaré was led to adopt, at least in a provisional sense, a definition of length congruence based on the propagation velocity of light (Poincaré 1906b:22). Soon afterward, Poincaré adapted his philosophy of geometry to the physics of electron theory. The Lorentz-FitzGerald “deformation”, Poincaré explained, had nothing to do with the principle of similitude: space remained identical to itself, while the material objects in it underwent deformations. Nonetheless, he continued, “we have no way of knowing if this deformation is real” (Poincaré 1907: 4). Lorentz’s theory of electrons, in Poincaré’s interpretation, presented no challenge to the relativity of space, and, at least at this early stage, required no modification of his position on the conventionality of geometry.

That same year, Minkowski introduced his geometric interpretation of the theory of relativity. It may be worthwhile to recall a few of the terms he chose to describe his theory.5454endnote: 54 The context of Minkowski’s Cologne lecture is discussed in Walter (1996).

We are accustomed,” Minkowski remarked to the scientists gathered in Cologne, “to look upon the axioms of geometry as finished with, when we feel ripe for the axioms of mechanics…”.5555endnote: 55 Minkowski 1909.

Apparently, the axioms of geometry were not as inviolable as Poincaré had allowed. The group of transformations preserving the form of the equations of mechanics, Minkowski continued, was

treated with disdain, so that we with untroubled minds may overcome the difficulty of never being able to decide, from physical phenomena, whether space [..] may not be in a state of uniform translation.5656endnote: 56 Ibid.

Minkowski went on to stipulate that the form of the laws of physics was invariant with respect to a different group than that of Newtonian mechanics, and to suggest that physics would find its most perfect expression in terms of relations between “world-lines” (Weltlinien) in four-dimensional space-time.

Couched in such terms, Minkowski’s theory was naturally viewed as a counter-example to Poincaré’s doctrine.5757endnote: 57 This was, by no means, the only response encountered. Geometric conventionalism was also discussed independently of the theory of relativity, for example, in Dingler & Dittrich 1911. With respect to Minkowski’s theory, Poincaré seems to have adopted a wait-and-see attitude. When he finally found the time to comment on it, in an essay appopriately titled “Space and Time”, the success of Minkowski’s formalism was well established; Poincaré noted the popularity of the four-dimensional space-time theory of relativity, to which he preferred the Lorentz electron theory, because the latter did not define space.5858endnote: 58 Poincaré 1912; Walter 1996.

As I mentioned in the introduction, Poincaré’s philosophy of geometry found no followers in Göttingen. Felix Klein, for example, had no use for any philosophy that advanced a demarcation between pure mathematics and the natural sciences. Geometry could be considered, according to Klein, as one of the physical sciences (Klein 1902).

Like Minkowski, Klein considered the theory of relativity as a domain particularly well-suited for development by mathematicians. Klein had a strong belief in the unity of the sciences, however, and in the mutual benefit to be gained from their intermingling. He followed Minkowski’s progress in relativity with great interest, introducing Minkowski’s four-dimensional formulation of electrodynamics into his lectures as early as 1907 (Walter 1996). Where Minkowski encouraged mathematicians to enter the field of theoretical physics (in as much as the theory of relativity was a part of physics), Klein expressed his hope that twentieth-century physicists would make use of the analytical tools developed by mathematicians in the nineteenth century (Klein 1910, Walter 1996).

Klein’s colleague David Hilbert once commented, sometime after Henri Poincaré perished, that while Poincaré was the most versatile and inventive mathematician of his time, in philosophical questions he had not been so fortunate.5959endnote: 59 Undated manuscript, NSUB, Hilbert Nachlass. This fragment may be tentatively dated ca. 1913, on the basis of a reference to Sommerfeld’s invited lectures. The absence of any mention of the general theory of relativity would constrain the date to sometime before the summer of 1915, when Einstein exposed his theory in Göttingen. The principle example of Poincaré’s unfortunate forays into philosophical questions, in Hilbert’s opinion, was his view on the role played by geometry in the knowledge of nature, which Poincaré himself would have called conventionalism. Hilbert recalled a lecture he had given the year before, in which he claimed that Poincaré’s doctrine was “in no sense justified at all.”6060endnote: 60 Undated manuscript, NSUB, Hilbert Nachlass. For a full translation of this document, see Rowe 1986: 76-7. My translation differs slightly from the one given by David Rowe. The basis for Hilbert’s judgment is unknown, although he argued in a 1918 lecture that, since the special theory of relativity ruled out the possibility of rigid bodies, the Euclidean structure of space as defended by Poincaré was consequently fictitious.6161endnote: 61 Majer 1995: 280. This view is similar to one expressed by Einstein, see Stachel (1980).

In Hilbert’s view, if Poincaré’s doctrine had gained some notable followers, this was because Poincaré’s immense authority had exercised an unwarranted influence. No names were mentioned, but Hilbert may have had in mind the philosophers identified by Eduard Study as laboring under Poincaré’s spell: Paul Natorp, Aloys Müller, Ernst Cassirer and Jonas Cohn. (Study 1914: 117).

Hilbert’s self-identification with the Göttingen tradition in mathematics and physics may be further judged by a remark made in his 1915 paper on the foundations of physics, containing the field equations of general relativity. Commenting on the historical significance of his theory, Hilbert recalled that Gauss had already invented a non-Euclidean physics, the invalidity of which was determined by measuring the angles of a mountaintop triangle (Hilbert 1924).6262endnote: 62 In celebration of Gauss’s supposed attempt to detect the non-Euclidean nature of space in the 1820’s, Felix Klein solicited contributions to build a monument commemorating the event (Bulletin of the American Mathematical Society 15 (1909): 318). In what was surely a happy coincidence, the planned inauguration of the Gauss Tower (on Gauss’s birthday) fell during the week of Poincaré’s visit in late April, 1909, on the occasion of the first Wolfskehl lectures. Poincaré accepted David Hilbert’s invitation to attend the inauguration (Poincaré to Hilbert, Hilbert Nachlass, NSUB).

Perhaps the most direct challenge to Poincaré’s doctrine came from the Cambridge scholar Alfred A. Robb. In the introduction to his essay entitled Optical Geometry of Motion, Robb opposed his view to that of Poincaré:

Speaking of the different “Geometries” which have been devised, Poincaré has gone so far as to say that: “one Geometry cannot be more true than another; it can only be more convenient.” In order to support this view it is pointed out that it is possible to construct a sort of dictionary by means of which we may pass from theorems in Euclidian Geometry to corresponding theorems in the Geometries of Lobatschefskij or Riemann. In reply to this; it must be remembered that the language of Geometry has a certain fairly well defined physical signification which in its essential features must be preserved if we are to avoid confusion. [..] It is the contention of the writer that the axioms of Geometry, with a few exceptions, may be regarded as the formal expression of certain Optical facts. The exceptions are a few axioms whose basis appears to be Logical rather than Physical. [Emphasis in the original.]

Robb, who had studied spectroscopy with Voigt in Göttingen, sought a physical foundation for geometry, in the tradition of von Helmholtz. Like most mathematicians, Robb acknowledged the validity of Poincaré’s argument of the intertranslatability of geometric terms between Euclidean and non-Euclidean geometry, although he underlined the drawback of the nominalist approach:

As regards the “dictionary,” we would venture to add that it would also be possible to construct one in which the ordinary uses of the words black and white were interchanged, but, in spite of this, the substitution of the word white for the word black is frequently taken as the very type of a falsehood.6363endnote: 63 Robb 1911: 1.

Contre Poincaré, Robb appealed to a somewhat vague notion of cognitive clarity, which may be understood as follows: geometry, be it Euclidean or non-Euclidean, had certain features which corresponded to experience, and which ought to be retained for the sake of clarity. On this basis, Robb proposed the definition–steadfastly refused by Poincaré–of geometric (and logical) relations by light signals.

The subtitle of Robb’s essay presented the optical geometry of motion as a “new view of the theory of relativity.”6464endnote: 64 The resulting geometry, metrically equivalent to Minkowski geometry, has been the subject of a number of studies, see Malament, Friedman. On the origin of his work Robb was not forthcoming, yet he considered that the formulas obtained agreed with those of Einstein, and he acknowledged Sommerfeld’s priority in considering–on the basis of “Minkowski’s theory”–the velocity of a particle as a hyperbolic tangent (1911: 30).

By the time Robb’s essay was published, Minkowskian relativity had gained a dominant position in the leading German physics journal, which was perhaps one reason why Robb offered no explicit justification of his rejection of Poincaré’s doctrine. Others, unlike Robb, explained why Poincaré’s view should be discarded.

For example, Felix Kottler, a mathematically-inclined theoretical physicist in Vienna, and the first to demonstrate that Maxwell’s equations were generally covariant, motivated his preference for physical geometry with an appeal to mathematical economy:

Poincaré’s point of view .. is to be contrasted with the law of mathematical economy. A physics in which light radiation propagates uniformly in a straight line and force-free material points move uniformly in a straight line is certainly mathematically simpler to handle (Kottler 1914).

In other words, from the point of view of mathematical simplicity, Kottler saw an advantage in considering light paths as straight lines (geodesics) in curved space, compared to a physics in which these paths were expressed in terms of curves in Euclidean space (as defended by Poincaré).6565endnote: 65 Kottler’s invocation of the “law of mathematical economy” was a clear reference to the view expressed by the Viennese philosopher Ernst Mach. Mach himself was one of the first to comment upon Minkowski’s space-time theory. His limited command of mathematics prevented him from following the more technical aspects of theoretical physics, of which Minkowski’s Grundgleichungen was a formidable example.

The argument of convenience was also advanced against Poincaré’s position by the Sydney mathematician Horatio Carslaw. While Carslaw did not consider Poincaré’s position to be discredited entirely by the theory of relativity, he maintained that it was “the Non-Euclidean Geometry of Bolyai and Lobatschewsky which, in some ways at least, is the more convenient.” Carslaw found an echo of Gauss in the writings of those developing the theory, and he cited Gauss’s remark to Taurinus that he would be glad if Euclidean geometry were not true, because “then we would have an absolute measure of length” (Carslaw 1916: 104).6666endnote: 66 The care taken by Carslaw to restrict his remark to certain aspects of the theory of relativity, the absence of any mention of Riemannian geometry, and his reference to absolute length measure all suggest that his object was the theory of relativity of Minkowski, and not the general theory of relativity.

The ascendancy of the point of view represented here by Kottler and Carslaw, that physics was simpler when expressed in non-Euclidean geometry, had been recognized–in part–by Poincaré, in one of his last essays (Poincaré 1912). As mentioned earlier, Poincaré noticed that some physicists had chosen a different convention concerning time, treating this as a fourth coordinate. The older convention still found its followers, Poincaré pointed out, predicting that this bi-conventional situation was liable to last for quite a while.

In fact, not all mathematicians considered Poincaré’s stance to have been contradicted by the theory of relativity. An author of several articles on non-Euclidean geometry, in addition to a comprehensive bibliography of this subject, Duncan Sommerville came to the conclusion that the principle of relativity was similar to Poincaré’s conventionalist stance on the Euclidean geometry of physical space. These were both impotence principles, according to Sommerville: conventionalism meant no experiment could establish that the geometry of space was not Euclidean, while the principle of relativity denied the possibility of establishing a privileged frame of reference (Sommerville 1914). A similar view was expressed by Poincaré’s nephew, the mathematician Pierre Boutroux, who recognized that his uncle’s positions had often stirred controversy, but that as far as the principle of relativity was concerned, this was just as conventional as geometry (1914).

Not just mathematicians, but some mathematical physicists also felt that the Lorentz-Poincaré electron theory was consistent with Poincaré’s philosophy of geometry. Théophile de Donder, a professor at the University of Brussels, shared his thoughts on recent progress in physics in a public lecture given in 1913.6767endnote: 67 De Donder 1913. A former student of Poincaré (see Glansdorff et al. 1987), de Donder was best known for his work on integral invariants (a research domain founded by Poincaré) and their applications in physics. In 1913, de Donder taught a course on the principle of relativity and its consequences, see L’Enseignement mathématique 15 (1913): 91. He evoked Poincaré’s distinction between geometry and mechanics: where mathematics was a “pure science”, the science of mechanics could never lose sight of its experimental foundations.6868endnote: 68 De Donder (1913): 11. Poincaré, we recall, had described the “new mechanics” brought about by the principle of relativity. As long as its foundations remained in mechanics–even a new mechanics in which mass varied with velocity–Poincaré’s doctrine seemed compatible with the theory of relativity.

Missing from de Donder’s account of electron theory and the principle of relativity was any mention of space-time, or of the geometrization of physics. As Poincaré himself had made clear, the two theories of relativity had different foundations: unlike the Minkowskian theory, the Lorentz-Poincaré electron theory did not involve a definition of space.6969endnote: 69 Poincaré 1912; Walter 1996.

## Concluding remarks and epilogue

In the first decade of the twentieth century, many professional mathematicians were familiar with non-Euclidean geometry, and interested in the foundations of geometry. Physicists, in comparison to mathematicians, were less concerned with questions of geometry. Henri Poincaré was an outstanding exception to such a predisposition among mathematicians towards a “physicalization” of geometry. Poincaré’s position on the conventionality of geometry was seen to be extreme, and was consequently rejected by mathematicians. This rejection reflected an overriding concern over the intellectual and institutional estrangement of mathematics from physics. The responses to Poincaré’s doctrine shed light on the way mathematicians viewed their discipline and its place among the sciences. These views, in turn, affected the direction in which mathematics and theoretical physics developed in the first decades of the twentieth century. A widespread concern for the physical relevance of mathematics should therefore be considered as a factor in the mathematical reception of both the Minkowskian theory of relativity, and the general theory of relativity of Einstein and Hilbert.

### Notes

• 2 Morris Kline (1990: 921-2) admirably exemplifies this position.
• 3 Studies I have consulted include Reichenbach; Grünbaum, Nagel; Sklar; Glymour; O’Gorman (1977); Torretti (1978, 1982); Giedymin (1982; 1991; 1992); Gillies (1986); Paty (1992); Heinzmann (1995).
• 4 Holton, Goldberg, Paty, Darrigol.
• 5 Walter 1996.
• 6 Among most active supporters of non-Euclidean geometry in France was the Bordeaux mathematician J. Hoüel, translator of Lobatchevsky, Riemann, Beltrami and Helmholtz, and partisan of a physical interpretation of non-Euclidean geometry (see Hoüel 1875).
• 7 Efforts to prove the parallel postulate did not end with that of Carton. Paul Barbarin recounted that a special Commission on Parallels was created by the Academy in order to examine the flood of demonstrations. Apparently, the Commission found no satisfactory submissions, since Carton’s proof had no successors in the Comptes Rendus.
• 8 Tannery 1876.
• 9 For a recent discussion of neo-Kantian responses to non-Euclidean geometry, see Panza 1995.
• 10 Some of Poincaré’s ideas on the foundations of geometry may be found in his earliest known manuscripts on automorphic functions, see Gray and Walter, eds., 1997).
• 11 Poincaré refined his ideas on the conventionality of geometry in the 1890’s, extending them to theoretical constructs in general. He also constructed arguments to support the tridimensional nature of space, while recognizing that this was dependent on the choice of basic elements (points instead of spheres, cf. Lie). His writings were collected and published in 1902 under the title La Science et l’Hypothèse. This volume went through several editions, and was translated into German and English by 1905, when a second volume of philosophical writings appeared, followed by others.
• 12 “…il existe dans la nature des corps remarquables qu’on appelle les solides et l’expérience nous apprend que les divers mouvements possibles de ces corps sont liés à fort peu près par les mêmes relations que les diverses opérations du groupe choisi.” Poincaré 1887:91. For reasons unknown, sometime between 1887 and 1891 Poincaré decided that solid objects did not exist in physical space. This decision may have led him to the more extreme position concerning the geometry of space, expressed in Poincaré 1891.
• 13 Helmholtz 1876.
• 14 Helmholtz was aware of the approximations inherent to measurement, see Helmholtz 1867: 372.
• 15 Precursors to Minkowski have been investigated in Bork (1964).
• 16 Boucher 1903; Jouffret 1903, 1906.
• 17 I thank Olivier Darrigol for pointing out Föppl’s earlier speculation on the existence of four-dimensional space, in the first edition of his treatise on Maxwell’s theory of electrodynamics (Föppl 1894).
• 18 Cf. Johann Bolyai, “The Science Absolute of Space,” ca. 1833, translated by Halsted, reprinted by Dover.
• 19 For a discussion of Helmholtz’s work on the foundations of geometry, see Richards (1977).
• 20 On the philosophical reception of Riemann’s lecture, see Nowak 1989. On Klein’s and Lie’s critiques of Helmholtz, see Richards 1977: 252.
• 21 Bonola 1912; Ziegler 1985; Lützen 1993. The relation of differential geometry and analytical mechanics in the late 19th century has been the object of studies by Lützen (1995), and Tazzioli (1993).
• 22 Transcriptions of letters to Poincaré from Hermite and Klein appear in the Cahiers du Seminaire d’Histoire des Mathématiques.
• 23 Course listings in the JDM-V indicate that in the first years of the 20th century, lectures on non-Euclidean geometry could be heard at Griefswald, Königsberg, Leipzig, Marburg and Münster. In England, Alfred North Whitehead taught this subject intermittently at Cambridge, starting in 1893 (June Barrow-Green, private communication), while in Paris no regular courses were given on non-Euclidean geometry per se (cf. the listings in L’Enseignement mathématique).
• 24 1890: 72. Klein lectured on non-Euclidean geometry in 1889, see Klein 1892.
• 25 Jacques Hadamard and Émile Picard were perhaps the most notable of these. Among the less eminent French mathematicians, opinion was divided. Lycée professor Paul Barbarin and Hermann Laurent (examiner at the Ecole Polytechnique) spoke out against Poincaré’s doctrine, yet Poincaré found ardent defenders in the Lycée professors J. Richard, and Adolphe Buhl (1911).
• 26 Barbarin (1902: 69ff); Laurent (1906: 40).
• 27 Enriques’s criticism of Poincaré’s conventionalist position on geometry was adopted by Eduard Study in his appeal for realism (1914).
• 28 Sommerville’s bibliography of non-Euclidean geometry (1911), the most complete of its kind, is one index of the relative interest of different fields and sub-fields in this domain, indicating also their chronological development.
• 29 See, for example, Coolidge (1909).
• 30 For a study of the Hilbert’s views on geometry which makes excellent use of this source, see Majer 1995.
• 31 Hilbert 1891: 6-7, in Majer’s translation (1995: 263).
• 32 On the emergence of theoretical physics in Göttingen see Stichweh 1984, and Jungnickel & McCormmach 1986. Klein’s relation to the Göttingen tradition is described at length by David Rowe (1989); for later developments see Mehra & Rechenberg 1982: 262.
• 33 Klein’s own productive period in mathematical research had also reached its end, as he no doubt recognized.
• 34 On the origins of this project, see Dyck (1898).
• 35 On the creation of the Göttingen institutes of applied mathematics see Rowe 1989. The strategic value of these institutes for advanced research in military technology was, perhaps, a consideration as important as the economic one, if not more so in the Wilhelmian era; Ludwig Prandtl’s aeronautics institute, for example, contributed to the advance of German aviation.
• 36 Rowe 1989.
• 37 Cahan 1986.
• 38 Jungnickel and McCormmach 1986.
• 39 David Rowe (1989:187) suggests that both Klein and Hilbert perceived a threat to their respective visions of the future of mathematics.
• 40 For the origins of the term Geisteswissenschaft, see Koehnke (1986).
• 41 A similar sentiment was expressed in France by Émile Picard (1911).
• 42 In 1906, Prussian universities had an enrollment of 1600 students in mathematics, a figure which climbed at a rate of six percent or more until at least 1911, by which time the entry of female students had pushed enrollment to over 2000. See Schönflies 1911: 27-8: PZ12: 311. Student enthusiasm for mathematics in Germany was further stimulated by the development of the theory of relativity (Max Abraham, 1912). A comparison with the enrollment trends in England may be instructive here, as Cambridge mathematicians were faced with declining student interest in mathematics, and increasing interest in the study of physics; see D.B. Wilson, HSPS. See also Warwick’s study of the development of numerical mathematics in England, and the establishment ca. 1912 of a laboratory of experimental mechanics in Edinburgh (1995). In France, attention was focused at this time on the decreasing number of foreign students in French universities, as more and more of these students preferred the German university system; see Hadamard, ca. 1905. The fact that a laboratory of experimental mechanics was also established at the Sorbonne by Gabriel Koenigs, ca. 1912, may suggest there was a renewed interest in applied mathematics in Paris at this time.
• 43 Anon., “Starke Überfüllung des höheren Lehrfaches mit Kandidaten der Mathematik,” Physikalische Zeitschrift 12 (1911): 311.
• 44 For a description of Minkowski’s contribution to the theory of relativity as an extension of the disciplinary frontier of mathematics, see Walter (1996).
• 45 Tannery 1903: 392.
• 46 The rejection of Poincaré’s doctrine was compatible with a more formal, abstract approach to mathematics, as may be seen from Hilbert’s example.
• 47 Comparisons of Einstein and Poincaré may be found in Miller 1981 and Paty 1992.
• 48 Planck 1908; Wien 1909.
• 49 The notation has been changed; Einstein first noted the mass-energy equivalence in 1905, and expressed this in 1907 with the formula $mu=E_{0}/V^{2}$ (1905:641; 1907a: 384). On Einstein’s later derivations of the gravitational deflection of light, see Earman and Glymour 1980: 55ff. The hypothesis was not exactly new–Newton conjectured in the Opticks that light was subject to the effects of gravitation. Closer to Einstein, Planck suggested in June, 1907, that if “latent energy” should gravitate, then inertial mass would be only slightly different from ponderable mass (CP2: 207)).
• 50 Gilbert Newton Lewis and Richard Tolman (1909:713); Einstein to Freudlich ca. 1911.
• 51 Federigo Enriques, for example, described Poincaré’s position on physical geometry as “overextended”, since there were physical hypotheses of a non-arbitrary nature (such as an influence of matter on the propagation of light) capable of verification, leading to the conclusion that the geometry of space was non-Euclidean (Enriques 1913: 36, 41). Indeed, with the general theory of relativity (1915), Poincaré’s view came to be understood as completely discredited in favor of that of Helmholtz. See Einstein 1921, 1926.
• 52 Poincaré did not entertain the possibility that the axis of rotation of a non-Euclidean solid coincide with a non-Euclidean geodesic described by the path of light.
• 53 Poincaré’s position remained consistent on this point, at least since he reversed his initial belief (Poincaré 1887) in the physical reality of solid objects in 1891.
• 54 The context of Minkowski’s Cologne lecture is discussed in Walter (1996).
• 55 Minkowski 1909.
• 56 Ibid.
• 57 This was, by no means, the only response encountered. Geometric conventionalism was also discussed independently of the theory of relativity, for example, in Dingler & Dittrich 1911.
• 58 Poincaré 1912; Walter 1996.
• 59 Undated manuscript, NSUB, Hilbert Nachlass. This fragment may be tentatively dated ca. 1913, on the basis of a reference to Sommerfeld’s invited lectures. The absence of any mention of the general theory of relativity would constrain the date to sometime before the summer of 1915, when Einstein exposed his theory in Göttingen.
• 60 Undated manuscript, NSUB, Hilbert Nachlass. For a full translation of this document, see Rowe 1986: 76-7. My translation differs slightly from the one given by David Rowe.
• 61 Majer 1995: 280. This view is similar to one expressed by Einstein, see Stachel (1980).
• 62 In celebration of Gauss’s supposed attempt to detect the non-Euclidean nature of space in the 1820’s, Felix Klein solicited contributions to build a monument commemorating the event (Bulletin of the American Mathematical Society 15 (1909): 318). In what was surely a happy coincidence, the planned inauguration of the Gauss Tower (on Gauss’s birthday) fell during the week of Poincaré’s visit in late April, 1909, on the occasion of the first Wolfskehl lectures. Poincaré accepted David Hilbert’s invitation to attend the inauguration (Poincaré to Hilbert, Hilbert Nachlass, NSUB).
• 63 Robb 1911: 1.
• 64 The resulting geometry, metrically equivalent to Minkowski geometry, has been the subject of a number of studies, see Malament, Friedman.
• 65 Kottler’s invocation of the “law of mathematical economy” was a clear reference to the view expressed by the Viennese philosopher Ernst Mach. Mach himself was one of the first to comment upon Minkowski’s space-time theory. His limited command of mathematics prevented him from following the more technical aspects of theoretical physics, of which Minkowski’s Grundgleichungen was a formidable example.
• 66 The care taken by Carslaw to restrict his remark to certain aspects of the theory of relativity, the absence of any mention of Riemannian geometry, and his reference to absolute length measure all suggest that his object was the theory of relativity of Minkowski, and not the general theory of relativity.
• 67 De Donder 1913. A former student of Poincaré (see Glansdorff et al. 1987), de Donder was best known for his work on integral invariants (a research domain founded by Poincaré) and their applications in physics. In 1913, de Donder taught a course on the principle of relativity and its consequences, see L’Enseignement mathématique 15 (1913): 91.
• 68 De Donder (1913): 11.
• 69 Poincaré 1912; Walter 1996.

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