Václav Hlavatý on intuition in Riemannian space

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Václav Hlavatý on intuition in Riemannian space Helena Durnová a,∗ , Tilman Sauer b a Masaryk University, Brno, Czechia b Johannes Gutenberg-Universität Mainz, Germany

Available online 4 June 2019

Abstract We present a historical commentary together with an English translation of a mathematical-philosophical paper by the Czech differential geometer and later proponent of a geometrized unified field theory Václav Hlavatý (1894–1969). The paper was published in 1924 at the height of interpretational debates about recent advancements in differential geometry triggered by the advent of Einstein’s general theory of relativity. In the paper he argued against a naive generalization of analogical reasoning valid for curves and surfaces in three-dimensional Euclidean space to the case of higher-dimensional curved Riemannian spaces. Instead, he claimed, the only secure ground to arrive at results is analytical calculation. We briefly discuss the biographical circumstances of the composition of the paper and characterize its publication venue the journal Ruch filosofický. We also give a discussion of the mathematical background for Hlavatý’s argument. © 2019 Elsevier Inc. All rights reserved. Souhrn Pˇredstavujeme anglický pˇreklad matematicko-filosofického cˇ lánku cˇ eského diferenciálního geometra a pozdˇeji jednoho z klíˇcových pˇredstavitel˚u geometrie pro teorii jednotného pole Václava Hlavatého (1894–1969), uvozený historickým komentáˇrem. P˚uvodní cˇ eský cˇ lánek vyšel v roce 1924, v dobˇe, kdy kulminovaly debaty o interpretaci nedávného pokroku diferenciální geometrie, stimulovaného Einsteinovou formulací obecné teorie relativity. Ve svém cˇ lánku Hlavatý horoval proti naivnímu zobecˇnování vztah˚u pro kˇrivky a plochy známých v trojdimenzionálním euklidovském prostoru na kˇrivky a plochy ve vícerozmˇerném zakˇriveném Riemannovˇe prostoru. Tvrdil, že jedinou bezpeˇcnou cestou ke spolehlivým výsledk˚um je analytický výpoˇcet. Pˇribližujeme také životopisné okolnosti vzniku cˇ lánku a cˇ asopis Ruch filosofický, v nˇemž cˇ lánek vyšel. Matematická tvrzení Hlavatého zasazujeme do širšího matematického kontextu. © 2019 Elsevier Inc. All rights reserved. MSC: 01A60; 53-03 Keywords: Riemannian geometry; Intuition in mathematics; Philosophy of mathematics; Mathematical communities; Václav Hlavatý

* Corresponding author.

E-mail address: [email protected] (H. Durnová). https://doi.org/10.1016/j.hm.2019.04.002 0315-0860/© 2019 Elsevier Inc. All rights reserved.

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1. Introduction This paper is a contribution to the history of differential geometry, taking as its starting point the reverberations that Einstein’s theory of relativity had in the mathematical community of differential geometers. It is also about the emergence of a community of differential geometers in the early twenties and the reaction of a young and ambitious mathematician on a newly emerging subfield of mathematics. We present an English translation of a paper, written in 1924 by the Czech mathematician Václav Hlavatý (1894–1969) and published in the Czech philosophical journal Ruch filosofický, entitled ‘On Intuition in Riemannian Space’. The paper had little impact over time and is not easy to understand, if only for the fact that it was written in Czech. Nevertheless, we believe that a closer examination of this paper and its historical context tells us something about the state of the art of differential geometry in the early twenties and about the influence that Einstein’s theory of general relativity had in the field of mathematics. We may also look at the paper as an example of how a young mathematician’s interaction with internationally renowned mathematicians in the early stage of his career urged him to share the cutting-edge research with his colleagues at home and whether he succeeded. By way of introduction, we will give some historical background to our translation of the paper and discuss the specific situation in which it was written. The paper is organized as follows. In Section 2, we will first provide some biographical information about Hlavatý and discuss his motivation to write his paper as well as the situation he found himself in at the start of his career, especially the Czech cultural context. We then introduce the journal Ruch filosofický as a publication venue in this context and give a little bit of background information about the academic situation in Czechoslovakia at the time in Section 3. In Section 4, we provide some linguistic-philosophical comments on the title of paper and discuss Hlavatý’s choice of terms. Section 5 gives a brief introduction and overview to the mathematical argument laid out in Hlavatý’s paper, and provides some background information necessary to understand his argument. We will finally attempt an assessment of the insights that can be gained by studying this paper. Our English translation of Hlavatý’s paper is given in an appendix. 2. Scientific community in Prague in the early 1920s and Václav Hlavatý’s early career Václav Hlavatý was born in Louny in the Austro-Hungarian Empire (now Czechia) on 27 January 1894.1 He graduated from the local Realgymnasium in 1913 and after studying engineering at Prague polytechnic for one year, he turned to mathematics and descriptive geometry. He served in the Great War and at its very end, in November 1918, was taken captive in Italy. He returned to Prague in the summer of 1919 to finish his studies. In April 1920, he began teaching in his hometown, Louny, at the school from which he graduated. He passed the teacher training examination in May of that year. He probably did this to earn a living, because already since 1919, he later claimed, he had been systematically studying modern differential geometry with the Prague geometer Bohumil Bydžovský (1880–1969).2 In 1921, Hlavatý submitted a doctoral thesis on the Plücker conoid to the faculty of natural sciences of Charles University in Prague. The thesis was reviewed in March 1921 by two Prague geometers, Jan Sobotka (1862–1931) and Karel Petr (1868–1950), and in June 1921 Hlavatý passed the doctoral exams in mathematics, theoretical physics (Hauptfach), and philosophy (Nebenfach). In mathematics and theoretical physics, Hlavatý’s performance was unanimously graded excellent, while in philosophy, the mathemati1 More detailed information about Hlavatý may be found in (Durnová et al., 2017). 2 Cf. Nožiˇcka (1969) and also Hlavatý’s application for a Rockefeller scholarship (Rockefeller Archive Center, collection In-

ternational Education Board, Series 1: Appropriations, Subseries 3: Fellowships in science, box 51, folder 793: Hlavatý, Václav (1927–1931)). Although Hlavatý would later claim that he had been studying differential geometry under the guidance of Bydžovský, the latter hardly worked in differential geometry. In comparison, the German University in Prague had Ludwig Berwald ˇ (1883–1942), and the young Masaryk University in Brno had Eduard Cech (1893–1960) on their faculties.

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cian Václav Láska (1862–1943) found Hlavatý’s performance excellent, as opposed to the opinion of the philosopher František Krejˇcí (1858–1934), who deemed it only sufficient. Although such discrepancy in the judgment of examiners need not be significant, this particular difference between Láska and Krejˇcí was indicative of a larger debate, as will be discussed in the next section. During his tenure as a teacher in Louny, Hlavatý was perhaps occasionally traveling to Prague to meet other mathematicians, or maybe he was only sending his mathematical papers to academic societies or journals for publication. In the summer of 1923, Hlavatý moved from Louny to Prague to start teaching at another Realgymnasium in September of that year. In the fall of 1923, he must have met the Dutch differential geometer Dirk Jan Struik (1894–2000), who got married in Prague on 14 July 1923. Upon returning home to Delft, Struik intimated to his teacher, Jan Arnoldus Schouten (1883–1971), that Hlavatý wished to study differential geometry with him. As a result, Hlavatý spent the summer semester of 1924 in Delft. He came to Delft in early February and stayed until early June. On 1 June, he gave a lecture on his research in Amsterdam and on 16 June, his work was presented at the French Academy of the Sciences in Paris by Jacques Hadamard (Hlavatý, 1924a). Hlavatý reported to Schouten from Louny about his return journey through Paris on 21 June 1924,3 and thus he probably composed and submitted his paper “On Intuition in Riemannian Space” to Ruch filosofický (Hlavatý, 1924b) after his arrival in Louny. It was published in the last issue of volume IV of Ruch filosofický, probably in December 1924.4 In the paper, Hlavatý referred also to his own paper, “Sur les courbes quasiasymptotiques” (Hlavatý, 1923–1924), which he had completed in Delft in May 1924 and submitted to the Dutch journal Christiaan Huygens.5 This was the paper Hlavatý intended to use as his habilitation thesis and he had announced this in his application of 1 October 1924,6 but since it was not printed in time, Hlavatý later changed his mind and instead designated one of his earlier papers (Hlavatý, 1923) as his habilitation thesis. Hlavatý’s habilitation was granted on 25 April 1925. Since our focus in this paper is on Hlavatý’s contribution to Ruch filosofický of 1924, we only summarize here in a few points his further career as a highly active and productive differential geometer: he was appointed professor at Charles University in Prague in 1931, teaching philosophy of mathematics as well as differential geometry until Czech universities were closed in November 1939. During the war, he could not teach, but stayed at home and worked on textbooks on differential geometry (Hlavatý, 1941a,b), which later also appeared in German and English translations (Hlavatý, 1945b, 1953), and on projective geometry (Hlavatý, 1944, 1945a). After World War II, he was briefly involved in politics and considered it impossible to stay in Czechoslovakia when the country was taken over by communist rule. He managed to emigrate to the United States with his family in the fall of 1948 and had a distinguished career at Indiana University in Bloomington. During these years, he became one of the most active researchers in unified field theory. His results appeared first in journal articles and then in a book (Hlavatý, 1958). He died in Indiana in January 1969.7 3 V. Hlavatý to J. A. Schouten, 21 June 1924. CWI Amsterdam, Schouten papers, folder 4 (1924). 4 Numbers 8 to 10 of volume IV(1924) of Ruch filosofický were issued together. The issue contained an obituary of Gustav Zába,

who died in October 1924, an article based on a lecture presented on 3 October 1924, and a report of a conference held in Berlin from 16 to 18 October 1924. Hlavatý’s paper does not contain information about when he submitted the paper; however, when he submitted his file for habilitation on 1 October 1924, he did not include “On Intuition in Riemannian Space” in his list of publications. Also, Hlavatý numbered his publications: the paper in Ruch bears no. 17. His no. 16 was presented at the Royal Bohemian Society for Science on 12 December 1924, his no. 18 was submitted in February 1925. 5 Hlavatý’s paper (Hlavatý, 1923–1924) appeared in the 1923–24 volume of Christiaan Huygens, but apparently this volume was only printed in 1925. 6 Archiv Univerzity Karlovy (Charles University Archive), Prague, collection Pˇrírodovˇedecká fakulta (Faculty of Science), personal file of Václav Hlavatý. 7 For more information about Hlavatý, see Durnová et al. (2017); Truesdell (1953); Lichnerowicz (1966).

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3. The journal Ruch filosofický Among the Czech philosophers in Prague around 1918, a doctrine of neopositivism was powerfully enforced by František Krejˇcí, who, as we saw, had been less than enthusiastic about Hlavatý’s graduate performance in philosophy. In 1918, coinciding with the founding of independent Czechoslovakia in October of the same year, František Krejˇcí became the unchallengeable leader of philosophy in Prague. Krejˇcí was ˇ the editor-in-chief of the main philosophical journal in Czech, Ceská mysl (Czech mind) and the only actively teaching professor of philosophy at Charles University. Philosophers of science therefore had a hard ˇ time getting published in Ceská mysl. Krejˇcí also enforced a more or less obvious requirement of explicitly referring to the philosopher and statesman, since 1918 the president of Czechoslovakia Tomáš Garrigue Masaryk (1850–1937), who was also a professor of philosophy at Charles University, in any piece of philosophical work. And other than one might expect from a philosopher with a penchant toward neopositivism, Krejˇcí showed little understanding for philosophy of mathematics and the sciences. Instead he held a true animosity against philosophy of the natural and exact sciences. The case of what Bydžovský called the first Czech philosopher of mathematics Karel Vorovka (1879–1929) demonstrates the tensions between the “pure” philosophers and the philosophers of science. Karel Vorovka8 was trained as a mathematician, but his teaching practice turned his attention to philosophy of mathematics. He was especially interested in the ideas of Henri Poincaré (1854–1912), whose views he explained and extended in his work On intuition in mathematics (O názoru v matematice) (Vorovka, 1917). His book was well received by the mathematicians (and physicists), who suggested Vorovka submit his book as a habilitation thesis. Bohumil Bydžovský called it the first work in philosophy of mathematics in Czech, and because of its considerable philosophical content, suggested an interdisciplinary committee be set up for the purpose of Vorovka’s habilitation. The physicist Bohumil Kuˇcera (1874–1921), the senior mathematician, geodesist, and polyhistor with keen interest in philosophy Václav Láska, and the philosopher František Krejˇcí were the members of this committee. Vorovka’s habilitation was approved in 1919, but Krejˇcí made sure that it was granted for philosophy of mathematics only.9 In 1920, the Faculty of Science in Prague emancipated from the Philosophical Faculty of Charles University on the basis of a decision taken in June 1920.10 Teaching in this new constellation started in winter semester 1920–1921. Philosophers of science formed their own section within the faculty of science, not within the remaining philosophical faculty; Krejˇcí kept his position at the philosophical faculty. From 1882 until the beginning of WWII, Prague had de facto two universities: one Czech and one German. Scientific communication in Prague in the first half of the 19th century would basically proceed in German. During the national revival in the 19th century, Czech emerged as a language of scientific communication. Still, the German University in Prague and the Charles University retained a fairly comparable structure. For example, both faculties of natural sciences were founded in 1920 and both had a chair for methodology and history of natural and exact sciences. These chairs were held by Karel Vorovka at the Czech and by Christian von Ehrenfels (1859–1932) at the German university, respectively.11 The growing dissatisfaction with Krejˇcí’s position led to the founding of the journal Ruch filosofický (Le Mouvement philosophique) in protest against the reigning neopositivist doctrine. Ruch was an important 8 On Karel Vorovka, see Pavlincová (2010). 9 Cf. (Pavlincová, 2010, pp. 84–86). 10 Similar developments may be observed also at other European universities, see, e.g., the near simultaneous emancipation of the

Science Faculties from the Humanities in Göttingen (1920), or Leipzig (1922). 11 By coincidence, von Ehrenfels retired in 1929, the same year as Vorovka’s death. Both chairs were only filled again in 1931: the Czech one by Václav Hlavatý, the German one by Rudolf Carnap (1891–1970). In 1929, the Vienna Circle Manifesto had been presented at the conference for the noetics of exact sciences, held in September 1929 in Prague.

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venue in this “fight for democracy in philosophy” (Pelikán, 1921). Its editors-in-chief were Karel Vorovka and Ferdinand Pelikán (1885–1952). Ruch’s first volume was published in October 1920, simultaneously with the emancipation of the Prague faculties of sciences from the philosophical faculties. It opened with Vorovka’s manifesto “Science and Philosophy” (Vorovka, 1921), in which Vorovka explained his general views on the role of science in philosophy and vice versa.12 The volume also carried an article on the philosophy of space by Václav Láska, a review by Vorovka of Oskar Kraus’s article Fiktion und Hypothese in der Einsteinschen Relativitätstheorie (Kraus, 1921),13 and a discussion on the principle of (special) relativity, which was continued also in later issues by the astronomer and Vorovka’s classmate Arnošt Dittrich (1878 –1959), by Hugo Szántó (1894–1974), and by Karel Vorovka himself. After 1926, the physicists František Nachtikal and Bohuslav Hostinský joined the debate. In short, Ruch became a venue for philosophically leaning articles on mathematics and physics. Articles were published exclusively in Czech, although German scholars also contributed to the journal (e.g. Otomar Pankraz (1903–1976), Hugo Szántó, Oskar Kraus (1872–1942)). It ran until World War II, the last issue appeared in 1941. 4. Linguistic discussion: názor vs. pˇredstava In the mid-1920s, a prominent problem for philosophers of science and also for Czech mathematicians of the time was the difficulty of visualizing the situations put forward by relativity theory (Jastrzembská, 2002). Into this setting, Hlavatý brought in an example where intuition might be misleading. The titles of Hlavatý’s paper and of Vorovka’s thesis were remarkably similar, we may hence assume that Vorovka had a particular interest in publishing Hlavatý’s paper: while Vorovka’s thesis dealt with intuition in mathematics in general, Hlavatý’s paper discussed the role of intuition in a special area, namely in the geometry of Riemannian spaces. The Czech word názor is a direct translation of the Kantian term Anschauung. The word carries direct visual connotations (schau in German, zor in Czech). The term is traditionally translated into English as intuition, whereby the visual connotations of the Czech and German terms (názor and Anschauung, respectively) is weakened despite the word’s etymology. The issue was picked up by Vorovka in his thesis. Vorovka explained in the introduction that, in accord with Poincaré, he would be using the extended meaning of intuition, which includes, alongside the primary meaning of intuition (názor) based in the senses, also logical intuition (názor logický). Vorovka set out to sketch how Poincaré’s scattered ideas would work if applied consequently to mathematics (Vorovka, 1917, p. 8). The topic was revived in 1924 by Václav Láska’s “On intuition a priori in mathematics” (Láska, 1924). He explained that the term názor had been found inadequate already by Bolzano, who found it confusing since it contains the element of vision (“zor”), making it difficult to talk of “logical názor (intuition)”. Láska nevertheless decided in favor of the term názor. Hlavatý in the second paragraph of his paper, introduced his main theme to be that of pˇredstava. That term also does not have an adequate English counterpart. In our translation we have left it untranslated. 12 Vorovka’s paper “Science and Philosophy” had been presented at the meeting of Jednota filosofická, the professional organiza-

tion of Czech philosophers, in March 1920. 13 Kraus’s standpoint is described by Vorovka as “mildly rejecting [zdrženlivˇe odmítavé stanovisko]”. On January 7 and 8, 1921, Einstein had delivered a lecture on relativity in an overcrowded great hall of Prague’s Urania where he engaged in a discussion also with Oskar Kraus. In correspondence, Einstein referred to this event as “very amusing” (“sehr amüsant”) (Einstein to Elsa Einstein, 8 January 1921, (Einstein, 2009, Doc. 12)). To his friend Paul Ehrenfest, he wrote: “With Kraus I had a public-discussion evening, an exceedingly amusing circus performance; but he was perfectly serious about it.” (“Mit dem Kraus hatte ich einen öffentlichen Diskussionsabend, eine überaus drollige Zirkusvorstellung; ihm wars aber ernst damit.”) (Einstein to Paul Ehrenfest, 20 January 1921 (Einstein, 2009, Doc. 24)). Einstein had been well known to Prague academic circles since his tenure as professor there at the German university from April 1911 till August 1912, see e.g. Kowalewski’s (1950, p. 237) personal recollection of Einstein’s apparently well received inaugural lecture.

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Quite literally, the Czech word “pˇredstava” corresponds to the German word Vorstellung and can roughly be translated as imagination, visualization, or perhaps even intuition. Neither term, however, seems adequate in this context. In the body of his paper, Hlavatý does not use the term názor, as it appears in the title, but rather opts for pˇredstava, which, compared to názor, refers more to imagination, which is what Hlavatý seems to be after in this paper: imagination deceives us, and pˇredstava therefore seems to be a better word than názor, suggesting seeing, which is difficult or impossible for higher-dimensional spaces. In other words, if we could see, we would not need to calculate, but we can only see two- or three-dimensional objects. The entire paper is about the possibility or lack thereof to transfer our sensual experience from a two- or three-dimensional Euclidean space to higher dimensions and curved spaces. 5. The argument of the paper Hlavatý starts out by referring to “Einstein’s gravitation theory” in its first three words, like a catch phrase. It is this one of the very few explicit published references to Albert Einstein (1879–1955) by Hlavatý in this early period. Many years later, in the 1950s, Hlavatý would embark on a long-term project to contribute to Einstein’s project of a unified field theory of gravitation and electromagnetism from the vantage point of a sophisticated modern understanding of differential geometry. In fact, Hlavatý emerged as one of the leading proponents of the mature unified field theory program (Goenner, 2014).14 Perhaps this later development was foreshadowed by the remark made at the end of this first paragraph where Hlavatý claimed that theoretical physics would no longer be an independent discipline but had instead become “part of another discipline, at first sight different, namely (differential) geometry.”15 The provocative, if not uncontroversial claim that physics had become part of differential geometry which appeared also in Hermann Weyl’s 1918 paper “Reine Infinitesimalgeometrie”16 gave a new dignity to the study of geometrical spaces, and more specifically Hlavatý’s paper addressed the widely discussed issue of the imaginability of higher-dimensional spaces, in particular of higher-dimensional curved spaces. With respect to the more modest challenge of extending notions of descriptive geometry, Hlavatý referred to work by Pieter Hendrik Schoute (1902),17 H. de Vries (1905),18 and himself (probably to his early works Hlavatý (1923b) and Hlavatý (1924c)). Klaus Volkert (2013) discussed the reception of non-Euclidean geometry in Germany in the second half of the 19th century, and in later work (Volkert, 2018) he discussed the gradual acceptance of the notion of a fourth dimension in the broader culture. In the context of interpreting relativity theory, an influential contribution would be Reichenbach’s Philosophie der Raum-Zeit-Lehre (1928), especially §§ 9–14, 44. While the question of Anschauung was discussed in the philosophical reception of relativity theory, it was most often done with respect to the pseudo-Riemannian 4-manifold of space-time, as e.g. in Minkowski space14 In fact, our initial motivation to take a closer look at Hlavatý’s 1924 paper came from the fact that it is one of the very few

explicit mentions of Einstein and his relativity theory in Hlavatý’s early years, see Durnová and Sauer (in preparation). 15 This very explicit interpretation of general relativity as a geometrization of physics was not Einstein’s own understanding, as was recently emphasized by Lehmkuhl (2014), see also the discussion in (Giovanelli, 2017, ch. 4). 16 “According to this theory, everything real that is there in the world, is a manifestation of the world metric; the physics notions are none other than the geometric ones.” (“Nach dieser Theorie ist alles Wirkliche, das in der Welt vorhanden ist, Manifestation der Weltmetrik; die physikalische Begriffe sind keine andern als die geometrischen.”) (Weyl, 1918, 385). 17 Pieter Hendrik Schoute (1846–1923) was professor of mathematics at the University of Groningen fom 1881 until his death. “Schoute was a typical geometer. In his early work he investigated quadrics, algebraic curves, complexes, and congruences in the spirit of nineteenth-century projective, metrical, and enumerative geometry. From 1891 he turned to geometry in Euclidean spaces of more than three dimensions, then a field in which little work had been done.” (Struik, 2017). 18 Hendrik de Vries (1867–1954) was Professor at the University of Amsterdam. He had studied projective and descriptive geometry with Otto Wilhelm Fiedler in Zürich, and then spent a few years as a mathematics teacher in Delft.

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time with Lorentz signature (−, +, +, +). With Hlavatý’s paper, we are concerned with generalizations of proper Euclidean or Riemannian spaces to higher-dimensional Riemannian manifolds. Hlavatý began with a terminological distinction between two meanings of the term “Riemannian space.” In a special sense, it refers to a non-Euclidean geometry of constant positive curvature. In a more general sense, it refers to a differentiable manifold with a variable metric field, given in terms of a quadratic differential line element. Hlavatý then summarized the current state of the art in differential geometry which, he said, now distinguished between 27 kinds of curved spaces.19 Hlavatý’s remark points to a significant difference between the conceptual developments in special and general relativity as a physical theory and as an application of advanced concepts of differential geometry. This development was to a large extent due to a change of conceptual view of differential geometry, namely a change of perspective from the metric to the connection as the pivotal concept. Einstein and Marcel Grossmann (1878–1936) had pioneered the use of the so-called absolute differential calculus (Ricci and Levi-Civita, 1901) in their search for a generally covariant gravitational field equation (Einstein and Grossmann, 1913). Their understanding as well as Einstein’s understanding of his final breakthrough to general relativity in late 1915 was entirely based on the notion of a differential line element determined by the components of a metric. Einstein did not originally recognize the role of the linear connection as an independent concept, a development that was initiated soon after by Tullio Levi-Civita (1873–1941), Jan Arnoldus Schouten, Hermann Weyl (1885–1955), and others (Reich, 1992; Cogliati, 2016; Cogliati and Mastrolia, 2018). Schouten presented the new theory of connections and of the kinds of curved spaces that could thus be discerned in Jena at the DMV meeting in 1921. From the point of view of differential geometry, however, the notion of a linear connection not only opened a new perspective on a geometric interpretation of the gravitational field equations and the associated concept of a Riemann curvature of four-dimensional space-time. The notion of a linear connection also opened a perspective that allowed the definition and classification of a very broad class of generalized Riemannian spaces. A classification of possible spaces on the basis of a general linear connection was given by Schouten (1922a). It distinguished 18 possible cases. In an addendum published a year later Schouten (1922b) added another 9 cases.20 In two further follow-up papers, Schouten (1923a,b) discussed the question whether generalizations of Riemannian geometry advanced by Blaschke and Reidemeister would fit into the general scheme, answering the question in the affirmative. A summary of the classification with 27 cases was later included in (Schouten, 1924, Pt.2, esp. §7) and also in (Berwald, 1923, § 31).21 A less general classification of affine connections was given by Cartan (1923). For further discussion, see Goenner (2004, sec. 5). In Schouten’s classification, he defined a covariant derivative ∇μ of tangent vectors v ν and linear forms vλ independently, thus he had 19 In fact, he talks about 29 kinds of curved spaces but we interpret the number 29 to be a misprint. There are several clear misprints

in the Czech text: In note 4 on p. 304 (here Note 32 on p. 73), the first index ω is μ in the original; on p. 308 (here p. 76), he misses a subscript and a closing parenthesis (‘V2 ) to Sn ’ instead of ‘V2 to Sn ’), on p. 309 (here p. 77) he misspells the name of Beez as Beer, and in the very last line of the paper Vμ−τ should probably be Vn−1 . We have silently corrected these misprints in our English translation of Hlavatý’s paper. On p. 303 (here p. 73) he expresses the condition of euclidicity in a fairly careless way. Curiously, also the reasoning on p. 305 (here p. 74) is obviously wrong, although Hlavatý himself must have known many counter-examples from relativity theory; for example, all Ricci-flat manifolds have vanishing scalar curvature. 20 About this classification, Schouten gave a report at the 1921 DMV-meeting held in Jena (Schouten, 1923a, p. 161). The meeting in Jena might have been of some significance for the development that is relevant here. At the same meeting, Ludwig Berwald as well as Wilhelm Blaschke and Kurt Reidemeister presented their ideas on Riemannian spaces (Berwald, 1922; Blaschke and Reidemeister, 1922). Dirk Jan Struik was present as well and, in fact, met his future wife Saly Ruth Ramler there (Rowe, 1994, p. 252). 21 Schouten presented his ideas also at the annual meeting of the DMV in Innsbruck on 25 September 1924 (Schouten, 1925). He also gave presentations on it in Münster and Göttingen in that same year.

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∂v ν + v λ νλμ ∂x μ ∂vλ ν ∇μ vλ = μ + vν  λμ ∂x

∇μ v ν =

67

(1) (2)

ν . For his classification scheme, he defined a tensor arising from the with 2n3 arbitrary coefficients νλμ , λμ distinction between differentiating covariant and contravariant quantities ν Cλμ ν = νλμ + λμ ,

(3)

a torsion tensor Sλμ ν =

 1 ν λμ − νμλ , 2

(4)

and a non-metricity tensor Qμ λν = ∇μ g λμ .

(5)

For his classification scheme, he then introduced vector quantities Cμ , Sμ , Qμ by the defining relations (using Einstein’s summation convention) Cμ = Cμν ν ,  1 Sλμ ν = Sλ δμ ν − Sμ δλ ν , 2 λν Qμ = Qμ g λν .

(6) (7) (8)

In the above, the first expression (6) stands for the trace of the sum of the covariant and contravariant connection, while the second and third equation (7), (8) describe special cases for quantities Sλμ ν and Qμ λν . They can both be either 0, or of the forms specified above, or in a general form. His classification scheme now derives from assuming independently for each of these three quantities whether either the full tensor Cλμ ν , Sλμ ν , Qμ λν is non-vanishing, only the vector quantity Cμ , Sμ , Qμ is non-vanishing, or both are vanishing. This gave him 33 = 27 possibilities. In his initial classification scheme, he had only considered the case of Cμ non-vanishing, which reduced the number of possibilities to 2 · 32 = 18 possibilities. Standard Riemannian geometry, relevant for Einstein’s theory of general relativity has all quantities vanishing. As an example of generalized geometries relevant for physical speculation, one may mention Weyl’s geometry which allowed for general non-metricity Qμ λν . Hlavatý mentioned that special cases of the general scheme involved Riemannian space, Weyl space, and affine curved space. In any case, the point of this reference to the 27 possible spaces seems to be able to justify the claim that theoretical physics became part of differential geometry, namely one very special case of those spaces, classical Riemannian geometry. Hlavatý went on to say that every traditional differential geometer uses pˇredstava in his work, as long as, we might add, they work with three-dimensional Euclidean space and its embedded curves and surfaces. He distinguished here between classical differential geometry, for which Klein’s 1872 Erlangen Program appeared to be his paradigmatic point of reference22 and modern differential geometry. The main ques22 In a footnote, he referred to work by his fellow countryman Eduard Cech ˇ (1893–1960), which he characterized as “differen-

tial projective geometry” and which he located at the boundary between classical and modern differential geometry. For further ˇ information on Cech, see e.g. (Katˇetov and Simon, 1993; Gray, 1994).

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tion arising for the latter was whether pˇredstava might play a heuristic role in the analysis of curved and higher-dimensional spaces. Specifically, the guiding question of Hlavatý’s article was: “Can pˇredstava of a Riemannian space be a regulative idea of mathematical operations pertaining to this space?” For further discussion of this question, Hlavatý then distinguished between, on the one hand, pˇredstava in its general meaning, which would be “used e.g. in the descriptive geometry of three-dimensional Euclidean space”. In contrast to this traditional meaning, he introduced what he called “logical pˇredstava”, which, he said, would be “the basis for the descriptive geometry of a higher-dimensional space.” Clearly, therefore pˇredstava has to do with geometric intuition, but since it can also have a logical component, it might refer to some kind of analogical reasoning as well. In any case, the main thrust of the article is to argue for the claim that in our analysis of higherdimensional curved spaces, any pˇredstava arising from Euclidean three-space is at best unreliable, if not misleading, and the only safe ground to move about in these realms is to do explicit calculations. Hlavatý added a disclaimer to the effect that the mathematical results he was using were not original but he did not give specific references at this point. In the following, we will comment on the argument and other details of his paper. As a preliminary remark, let us state that reading Hlavatý’s text requires some attentiveness, since in the course of the exposition, Hlavatý moved from a 2-dimensional space (a surface) to 3- or more-dimensional space (a space) and back several times, without an extra warning to the reader. This approach is curious, considering Hlavatý wished to reach out to an audience without any prior training in geometry. For further discussion of Hlavatý’s argument, it will be useful to introduce some notation that was widely used in the literature on differential geometry at the time (see e.g. reviews by Struik (1922); Berwald (1923); Schouten (1924)) but largely fell out of usage. Hlavatý only introduced it late in his paper. He may have considered these terms and notation common knowledge, but on the other hand, the paper was published in Ruch filosofický and thus addressed a general audience. Introducing half a dozen of technical terms only after he had used it made the paper hard to read for his contemporaries. As for notation, Hlavatý denoted by Vn a general Riemannian manifold, i.e. in more modern terms, an n-dimensional topological space with a differentiability structure and a metric field gμν compatible with variable curvature. More specifically, he used Sn to denote an n-dimensional Riemannian space of constant (positive or negative) curvature and Rn an n-dimensional Euclidean space, i.e. an n-dimensional Riemannian space with zero curvature. In other words, Sn could be a (hyper)sphere or a hyperbolic space. His original question then divides up into two questions, namely whether pˇredstava can be used to generalize from n-dimensional Euclidean space to n-dimensional Riemannian space, ?

Rn −→ Vn and, second, whether it can be used to generalize from a curved two-dimensional space, i.e. a surface, to higher-dimensional Riemannian spaces, ?

V2 −→ Vn

(n ≥ 3).

Before discussing his examples in more explicit detail, Hlavatý restricted himself to the special case of Riemannian spaces proper which he characterized as metric spaces with a linear connection that is angleand length-preserving as well as torsion-free. About the four conditions characterizing Riemannian spaces which he expressly stated, one may note that in the first condition, he introduced the metric tensor with its contravariant components, as was done as well, e.g., in (Struik, 1922, p. 25), rather than following the perhaps more natural and more common way of introducing the metric via the line element and its covariant components, as done e.g. in (Berwald, 1923, p. 127). The second, third, and fourth condition together amount to a statement of the fundamental theorem of Riemannian geometry about a uniquely

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defined metric-compatible and torsion-free Levi-Civita connection. For these spaces, he introduced what he called the “Riemann-Christoffel-Affinor” Kωμλν , which is just the general Riemann curvature tensor and he also said he would refer to it as the tensor of curvature.23 In order to explain the meaning of the curvature tensor, Hlavatý invoked the case of two-dimensional curved surfaces and explained the meaning of curvature in terms of the Gauss mapping. This explanation, it appears, was directed at an audience that had been exposed to some advanced or at least pertinent mathematical training. One may observe again that this counteracts the apparent motivation of the paper. Based on the notion of Gaussian intrinsic curvature for surfaces, Hlavatý proceeded to characterize curvature of higher-dimensional spaces in terms of the notion of sectional curvature, i.e. of the Gaussian curvature of surfaces formed by geodesic curves through a point P tangent to a given plane through P . Hlavatý did not actually use that term, but described the notion on p. 74, in the second paragraph below Eq. (A). Hlavatý’s first observation was that in the case of Euclidean spaces, characterized by a vanishing of the curvature tensor24 Kωμλν = 0,

(9)

the Euclidean character can be generalized, without problems, to higher dimensions. He then stipulated, without further justification, that “in order to be able to extend correctly the logical pˇredstava from a [Riemannian] surface to a Riemannian space of at least three dimensions, it is necessary that both spaces have the same structure (as it was in the previous case), i.e. that their tensors of curvature are of the same form, independent of dimension.” Hlavatý did not give an argument for this claim, nor did he define in any way what he meant by “of the same form”. He would have been aware, we believe, of the problematic that the curvature does not, in general, determine the metric.25 What he may have had in mind becomes only a little clearer in his further discussion, where he introduces spaces of constant curvature, K ≡ const. These have curvature at each point of the form K(gωλ gμν − gμλ gων ),

(10)

irrespective of the dimension of the space. Here K(x) = K is the two-dimensional Gaussian curvature of each section, a constant that neither depends on the direction of the section nor on the point of the manifold. In this context, Hlavatý invoked a well-known, often-cited theorem due to Friedrich Schur (1856–1932) who had proved in 1886 that if the curvature of an connected Riemannian manifold of dimension n ≥ 3 is written in the form (10) with general sectional curvature K(x) and if K(x) does not depend on the sectional direction at any one point then is also constant at every other point, K(x) = K = const.26 23 The terminology of “Riemann-Christoffelscher Affinor” for the Riemann curvature tensor and of affinors in general for ten-

sors was explicitly used, e.g. by Struik (1922), who also gave a reason for calling it this way. He wrote: “Die kontravarianten, kovarianten und gemischten Affinoren heißen bei Ricci, bzw. Einstein kontra-, kogrediente und gemischte Systeme bzw. Tensoren. F. Jung verwendet den Namen Affinor, weil mit jedem Affinor zweiten Grades mit nicht verschwindender Determinante eine affine Transformation korrespondiert [...]. Es existiert aber noch ein viel wichtigerer Grund diesen Namen zu verwenden. Sind doch die Affinoren verschiedenen Grades gerade die einfachsten zur Gruppe der affinen Transformationen gehörigen geometrischen Größen. Der aus der Elastizitätstheorie stammende Name Tensor ist seinem Ursprunge gemäß nur für symmetrische Größen zu verwenden.” (Struik, 1922, pp. 18–19). Another indication of the influence of the Schouten-Struik tradition on Hlavatý is found in his footnote on the curvature tensor where he refers to the techniques of so-called “symbolic method” using “ideal vectors” which feature prominently in Schouten’s and Struik’s works from these years. 24 For a recent historical analysis of the role of Riemann’s work, and of the tensor bearing his name, in the history of differential geometry, see Cogliati (2014). 25 For a modern discussion of this point, see (Berger, 2003, sec. 4.5, pp. 213–216). 26 For a more modern formulation and proof of Schur’s theorem, see, e.g., (Kobayashi and Nomizu, 1963, p. 202).

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Hlavatý then discussed two examples that are meant to demonstrate his claim that pˇredstava fails in guiding us to generalize from surfaces or Euclidean three-spaces to higher-dimensional Riemannian spaces. In his first example, he looked at a two-dimensional submanifold V2 in an n-dimensional Riemannian manifold Vn . That V2 is constructed by looking at all geodesics in Vn passing through a point P ∈ Vn whose tangent vector is in the same Euclidean plane. The metric in Vn induces a metric in V2 and, by construction, any curve geodesic in V2 that passes through P is also geodesic in Vn . The question then is whether V2 is a totally geodesic submanifold, i.e. whether all geodesics in V2 are also geodesics in Vn . The answer is that this is only the case if Vn has constant curvature, i.e. is actually an Sn . The meaning of this first example can be summarized as follows. According to Hlavatý, geometers would expect that the usual intuition that works in Euclidean space would work for curved spaces as well: one expects that formulas in n-dimensional Euclidean space Rn , where n > 3 work the same as in 2- and 3-dimensional spaces. But for curved spaces this is only true in two other cases: Lobachevski and the special kind of a Riemannian space with positive constant curvature. The second example concerns a mathematical result of Hlavatý’s that he had just recently found and apparently was proud of. As he emphasizes, the motivation for writing this philosophical reflection in Ruch really comes from that technical mathematical result he had obtained earlier and published in his almost simultaneous paper in Christiaan Huygens (Hlavatý, 1923–1924). In the Ruch paper, the problem was stated as follows: Example two. Let us consider a surface V2 in a space Vn . Let us put an arbitrary n − 1-dimensional space containing a tangent direction of this surface in its point B. Let us extend all directions in this space from point B geodesically and let us find the intersectional curve of the thus space Vn−1 obtained in this way with the surface V2 . We now have to decide whether all normal lines to that curve in point B are contained in Vn−1 .

Hlavatý’s example was expressed only in a loose, non-technical way here using Czech language that was partly introduced originally for this purpose as there was no established tradition of Czech texts treating advanced differential geometry. It is not quite clear, therefore, how to translate it precisely into modern, or even contemporary terms. As his own reference indicates, the example refers to the highly technical and almost simultaneous paper on so-called quasi-asymptotic curves that he had recently published in Christian Huygens (Hlavatý (1924a), see the discussion above in Section 2). That result was developed and discussed in the idiosyncratic technical framework and notation used by Schouten and Struik at the time. It engaged with a notion of quasi-asymptotic curves introduced by the Italian geometer Enrico Bompiani (1889–1975) that, to our knowledge did not play any role in the later literature on differential geometry.27 We refrain from attempting to give an assessment of this work from a modern point of view but only observe that, according to Hlavatý, the answer to his question, again and also somewhat counterintuitively, is positive only if the embedding manifold is of constant curvature, in Hlavaty’s words, if the “Vn is of the special form Sn ,” i.e. a (hyper)sphere or a hyperbolic space (Hlavatý, 1924b). In the final paragraph, Hlavatý referred to a mathematician Richard Beez (1827–1902) who proved the theorem (1875a) that an n − 1-dimensional Riemannian manifold that is embedded in an n-dimensional Euclidean space, in general, will not be deformable. About Beez’s work (1874; 1875a; 1875b; 1876; 1879), Struik (1922, p. 2) remarked that his attempts to demonstrate the inconsistency of the Riemannian theory of the Vn failed but led him to the important theorem, that shows a curious exceptional role of the cases n = 2 and n = 3 for the deformability of a Vn−1 in an Euclidean space of dimension n. Later, Killing (1885, pp. 263–4) added critical remarks about exceptions from Beez’s original theorem, which would invalidate his reservations against the notion of higher dimensional manifolds. Beez (1888) elaborated on 27 There is a brief discussion of this notion in footnote 309 of (Berwald, 1923, p.153).

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his criticism, for further discussion, see Volkert (2013), and, for a modern formulation and proof of rigidity theorems due to Beez and Killing, see Kobayashi and Nomizu (1969, pp. 45–46). 6. Concluding remarks Given the fact that almost all of Hlavatý’s papers were published in mathematical journals, we might ask why he published his paper “On Intuition in Riemannian Space” in Ruch filosofický, a philosophical journal. He may have been simply trying to reach out to a wider audience, making use of a recent technical and non-trivial result of his that he had achieved in the field. It also so happened that, for Hlavatý, the paper “On Intuition in Riemannian Space” later had most welcome, if perhaps unintended and unforeseen consequences. Having written this paper for a philosophical journal was a key point in the case made by Prague Czech mathematics professors when they suggested Hlavatý should be offered the chair for philosophy of mathematics after Karel Vorovka’s premature death in January 1929. It was decided that Hlavatý’s teaching obligations would be divided between philosophy of mathematics and geometry, which suggests that, in fact, the mathematics professors in Prague also wished to appoint a differential geometer. When writing the paper, Hlavatý was a self-confident young mathematician whose career, he may have felt, was delayed by his service in the army during WWI. He also may have observed a notable difference between the styles of mathematics done in Delft and in Prague, and this would hold even more for philosophy of mathematics. With his other Czech writings from this period he was trying to inform other mathematicians and philosophers in Prague about recent results in his field of expertise. In the Ruch paper, he also challenged a number theorist to join him in explaining the pitfalls of number theory in a similar vein, but nobody seems to have answered to this challenge. Hlavatý did not mention anybody explicitly, but he could have meant his peer Vojtˇech Jarník (1897–1970). With regard to philosophy, Hlavatý might have wanted to join the initiative by Vorovka and his colleagues to fight against the predominance of Krejˇcí, but he seems to have lost interest when nobody really reacted to his paper. Hlavatý was writing this paper for his Czech colleagues but just as it would be Hlavatý’s only philosophical paper it also remained the only contribution to Ruch with technical mathematical content. There were other papers on mathematics and logic, but all of these were rather about mathematics. Hlavatý intended his paper to be a contribution to the debate on intuition and an invitation to other mathematicians to join the debate. It follows the tradition of translating mathematical language into Czech, so as to create a technical vocabulary. This tradition began in the mid-19th century during the Czech national revival and aimed at the revival of a sophisticated language. But in the case of Hlavatý’s paper, it seems as if the Czech community lacked the critical mass to continue this discussion in Czech. The paper finally documents the reverberations of Einstein’s new theory of general relativity into the community of differential geometry in the twenties, and through this reception by geometers into a broader cultural discourse. But the mathematical interpretation that the issue of intuition gained in this debate did not help in transmitting the debate to a broader debate in philosophy. Hlavatý’s main argument that intuition inevitably fails in higher-dimensional general Riemannian spaces remains a valid standpoint and it is given non-trivial and little known justification in this paper. Acknowledgments We wish to thank G. Alberts, J. Kot˚ulek, D. Rowe, E. Scholz, R. Siegmund-Schultze and V. Žádnik as well as the referees for helpful comments on earlier versions of this paper. ˇ grant no. 15-11070S, Mathematics, physics, and politics: life and work of This work was supported by GACR Václav Hlavatý (1894–1969) in international context.

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Appendix. V. Hlavatý, On intuition in Riemannian space (1924)28 Einstein’s gravitation theory turned our attention to non-Euclidean spaces. In classical geometry, this notion was reserved for very special metric spaces, namely for Riemannian space in the narrower sense of the word and the Lobachevsky space. (Both will be discussed later on.) Contemporary means allow us to distinguish altogether 27 different kinds of curved spaces. The simplest of them is the general Riemannian space, which contains both spaces just mentioned as well as Euclidean space as special cases. Einstein’s theory has been fully developed in Riemannian space, Weyl space, and affine curved space. Through the influence of Einstein’s theory, theoretical physics ceases to be an independent discipline and becomes part of another discipline, at first sight different, namely (differential) geometry. For a classical differential geometer, who usually worked in Euclidean space, pˇredstava was an almost indispensable tool. In general, we can say that every geometry that followed Klein’s program (“Erlanger Programm” from 1872) was inspired directly by the pˇredstava.29 It is thus natural that a question arises whether and to what extent it is possible to permit the pˇredstava in modern differential geometry, dealing with curved spaces. This question may interest a mathematician as much as a theoretical physicist, with regard to the above-mentioned coordination of physics and geometry. In this article, I wish to restrict myself to Riemannian space and answer the following question: Can the pˇredstava of a Riemannian space be a regulative idea of mathematical operations pertaining to this space? In this paper, I understand the word “pˇredstava” both in its general meaning, used e.g. in descriptive geometry of a three-dimensional Euclidean space, and in the sense of “logical” pˇredstava, which is the basis of descriptive geometry of a higher-dimensional space. After negatively answering the question posed, I will introduce two examples from my own practice which clearly show the divergence of the pˇredstava from the calculation. ∗ The problem posed can be divided into two questions: 1. To decide whether the kind of pˇredstava is permitted that draws conclusions for Riemannian space based on Euclidean space.30 2. To decide whether the kind of pˇredstava is permitted that draws conclusions for Riemannian space on the basis of the situation in a surface (through adding dimensions). In order to be able to answer both of these questions, we will first define the space under consideration. On the basis of this definition, we will reveal the internal structure of that space. (By this I mean those of its properties that are intrinsic to it, i.e. properties related to its curvature). If the sought for structure will have the same properties as the structure of a surface or of a Euclidean space, then we will be able to speak of the pˇredstava as being permitted in both cases suggested.—In order to do this, I will use mathematical results that are not original. I will be leaving out calculations as much as possible. 28 [Hlavatý, Václav, 1924, “On názoru v prostoru Riemannovˇe”. Ruch filosofický, vol. 4, pp. 302–309. Translated by Helena

Durnová and Tilman Sauer. We leave the term pˇredstava, roughly meaning vizualization, imagination, or intuition, untranslated throughout the text but render it italicized. For a commentary on its meaning, see p. 64 above. A couple of misprints in the original Czech version have been silently corrected, see Note 19 above. All following footnotes in this appendix are Hlavatý’s own footnotes; the references following the appendix are to our historical commentary.] 29 There are of course exceptions. In my opinion, one such exception is differential projective geometry (cultivated in this country ˇ by Professor Cech). But this geometry is almost on the boundary between Klein’s definition and the modern understanding. 30 Unless noted otherwise, I will use the word space in all cases when more than two dimensions are involved. As is usually done, I shall call a two-dimensional space a surface.

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Riemannian n-dimensional space (n > 2) is characterized by the following properties: 1. The space is metric. This means that in any of its points, it is possible to measure lengths and angles. 2 We do so by means of a set of values that are determined by g μλ = g λμ (there are in total n 2+n of those values; in Euclidean space, they are all equal to 1). 2. The space is conformal. We call a space conformal if the angle of two directions is preserved in any parallel transport. 3. The metric of the space is constant. This means that in any parallel transport (in the non-Euclidean sense), the length of a vector does not change. 4. The space is symmetric. This property—the name of which is chosen in a somewhat unfortunate way as a result of the relevant mathematical expression—is geometrically interpreted in the following way: If we construct an infinitesimal parallelogram, the two sides of which are formed by two infinitesimal transports of a point, then such a parallelogram is a closed curve. Any space satisfying these conditions is a Riemannian space.31 It is immediately clear that surfaces in the usual sense of the word and Euclidean spaces also belong to the class of Riemannian spaces. It hence follows that the internal structure of these spaces must in general be the same. Only a specialization of that structure will allow us to distinguish between the different cases, as we will now show. If we were to express all four conditions stated above in a mathematical way, we would see that they all depend on gλμ . It is therefore likely that the structure of the space will depend also on gλμ , which is the only quantity which characterizes this space. As the structure should be intrinsic to the space, it is obvious that it must not depend on the choice of the values determining the space, the so-called parameters. The so-called Riemann-Christoffel affinor satisfies both conditions. It is a set of quantities that I will denote as Kωμλν 32 (ω, μ, λ, ν = 1 . . . , n). We will call this set the tensor of curvature. The tensor of curvature can be interpreted geometrically in multiple ways. I will discuss here the one that will help us solve our problem. Let us consider a surface in a three-dimensional Euclidean space. Let us choose an arbitrary point B on this surface and trace an infinitesimal curve on the surface around this point. Let this curve delineate an infinitesimal element of the surface with area . Let us put lines that are parallel to the normal lines at each point of the infinitesimal loop through an arbitrarily chosen centre of a sphere with radius equal to one. Their intersections with the sphere will determine an infinitesimal loop ∇ with area ∇ on the sphere. The ratio is called Gaussian curvature of the surface in that point. λμ If for numbers g we determine numbers gλμ such that  λ gλμ g

λν

=

1 for μ = ν , 0 for μ = ν

then the Gaussian curvature will be determined by the formula 31 As an interesting aside, let me add that replacing conditions 1 and 3 by the following condition:

α) Only right angles are defined, or β) not even right angles are defined, together with conditions 2 and 4 leads to Weyl space, or to affine space, respectively. 32 If we denote by a such ideal values that a a = g , and ∇ μ is the symbolic expression for derivation in Riemannian space, μ λ μ λμ then Kωμλν is determined by the well-known equation 2(∇[ω∇μ]aλ )aν = Kωμλν , which clearly shows its dependence on gλν .

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ω,μ,λ,ν g ωλ g μν Kωμλν .

(A)

Let us consider a tangential plane (which is a geodesic surface in the three-dimensional Euclidean space) through point B of the surface under consideration. Let us put two orthogonal directions through point B, denoted by B1, B2. If we consider these two directions to be the coordinate axes, then the only existing component of the tensor of curvature of these axes will have exactly the form of the sum just stated. For the sake of brevity, let us denote it by K1212 . The Gaussian curvature of the surface is thus K1212 . In general, it varies from point to point. Let us now put n orthogonal directions through a point B of an n-dimensional Riemannian space and denote them by B1, B2, . . . , Bi, . . . Bn. Let us put through any two directions Bi Bk a surface which is geodesic in point B. By mathematical operations, similar to the ones suggested above, we will find the Gaussian curvature of this surface Kikik . That [curvature] is in general different from zero. If we determine the Gaussian curvature of all n(n−1) surfaces that can be put through point B in a Riemannian 2  space in the way just explained, and if we add all the curvatures thus obtained, we will get the formula ik Kikik which is equal (up to a constant factor) to the expression A). This formula is called the Gaussian mean curvature of a Riemannian space and is in general different from zero. Should it be equal to zero, it is necessary that each of the expressions Kikik is equal to zero, Kωμλν = 0. Such a space is a Euclidean space: a Riemannian space whose tensor of curvature is equal to zero is a Euclidean space. Notice that the dimension of the space is irrelevant, as long as it is greater than one. For that reason, we can create a (logical) pˇredstava an n-dimensional Euclidean space (n > 3) based on three-dimensional space. The correctness of this theorem is confirmed also by the existence of descriptive geometries of higher-dimensional Euclidean spaces.33 We now come to the core of the problem: If we can create by a (logical) pˇredstava a higher-dimensional Euclidean space because its curvature tensor (as that of a plane or as that of the space we live in) is equal to zero, can we then perhaps also create by a (logical) pˇredstava a general Riemannian space on the basis of a surface with non-vanishing curvature tensor? In order to be able to extend correctly the logical pˇredstava from a surface to a Riemannian space of at least three dimensions, it is necessary that both spaces have the same structure (as it was in the previous case), i.e. that their tensors of curvature are of the same form, independent of dimension. Let us then examine first the curvature tensor of a surface. If gλμ are again quantities describing it, it can be proved easily that the curvature tensor must be of the form K(gωλ gμν − gμλ gων ),

(B)

where K (earlier denoted as component K1212 ) is the Gaussian surface curvature, in general different at each point. In order to get the logical pˇredstava of a Riemannian space, starting from a surface, the curvature tensor of that space would have to be of form (B). In such a space, the Gaussian curvature of any surface geodesic in point B is the same and independent of the particular choice of directions Bi, Bk. This, however, does not hold in the general case. This has an important consequence. The logical pˇredstava of a general Riemannian space based on a surface is impossible. It is, however, natural to look at spaces whose curvature tensor is of form B and which have the abovementioned geometrical properties. In 1886, Schur proved that the tensor of curvature which is of the form 33 Works of P. H. Schoute, H. de Vries, and the author.

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(B) must also necessarily have a constant K (greater or smaller than zero). The sought for spaces are special Riemannian spaces whose Gaussian scalar curvature is constant. We distinguish three types of these spaces: 1. Lobachevskyan space (K < 0), 2. Euclidean space (K = 0), 3. Special Riemannian space (K > 0). The tensors of curvature of a surface and of any of these three spaces have the same form, irrespective of their dimension. It follows that we can get a logical pˇredstava of special spaces 1 and 3 (the case of Euclidean space has already been dealt with) based on a surface. As far as a generalized Riemannian space is concerned, a logical pˇredstava based on Euclidean space is obviously not allowed, since the structures of the two spaces are different. We can thus summarize the results obtained in the following proposition: A (logical) pˇredstava of a higher-dimensional Riemannian space, based on the image of a surface, is not possible in general. But there are special Riemannian spaces (Riemannian space in the narrow sense of the word, Euclidean, and Lobachevskyan space), for which we may create a logical pˇredstava based on the pˇredstava of a surface or a three-dimensional Euclidean space. We must, however, be careful when using this proposition, since there are apparent exceptions. Examining them more closely, however, we will find that these apparent exceptions only confirm the rule. I will discuss an example. · · · ν defined The structure of space is given by the expressions Kωμλν . If we now consider quantities Kωμλ by the equation ···ν Kωμλ = α g να Kωμλα , · · · ν also characterize the structure. If therefore these expressions we will see easily that the expressions Kωμλ are different for two spaces, or for a surface and a space, then according to the above proposition, it is not permitted to generalize from one space (surface) to the other. · · · ν are different, the expressions But if the components Kωμλ ···λ λ Kωμλ

may be the same for the two spaces (the surface and the space). It follows that properties of the space that depend only on the latter expression can be common to both spaces, even if their tensors of curvature were different. It is precisely the equality of both expressions derived from the curvature tensors that would allow us to transfer conclusions from one space to another. But since we can only assess the logical pˇredstava on the basis of the curvature tensor, our proposition remains valid. ∗ In the following, I will give two examples that clearly demonstrate the impossibility of a logical pˇredstava of a Riemannian space. I draw attention especially to the second example, where the image not only does not seem to leave us in doubt, but it also leads to an incorrect conclusion. For the sake of brevity, I will denote n-dimensional Riemannian space by Vn , special curved space by Sn , and Euclidean space by Rn . Example one. Take two arbitrary directions through a point B of a space Vn and put a surface V2 through them. Let us assume that this surface is made of curves going through B geodesic in Vn (there are exactly

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∞1 of them). This surface is certainly geodesic in B in the space Vn . We have to determine, however, whether it is also geodesic in its other points? Let us first consider the case where Vn is Euclidean space, i.e. Rn . Then the surface V2 is a plane (R2 ) which is geodesic not only in point B, but also in all its other points. If, then, we made conclusions from Rn to Vn , we would answer the question positively. But we have shown that Rn is not curved, while Vn is curved. It is thus not strange that a positive answer would be incorrect. We can, however, help ourselves by using another space, the only curved space that is not outside of our imagination, namely the surface V2 . We will first perform the task on a surface and then we will move to Vn by a logical pˇredstava. How would the above-stated question be posed? “We will choose a direction in the surface V2 and geodesically extend it. Is the curve created in this way a geodesic curve on V2 ?” For a mathematician, such a question is a tautology. He will immediately answer it positively. If he logically extended in this way statements from V2 to Vn in the same way as we extend statements from a plane to R3 or from R3 to R4 , etc., his answer would be the same as in the first case. On the basis of what we have stated before, the positive answer is only correct when special spaces Sn and Rn are used, which have the same structure as the plane. The calculation indeed shows that in Sn (or Rn ), the above-mentioned surface is geodesic in each of its points (and thus it is S2 or R2 ). Example two. Let us consider a surface V2 in a space Vn . Let us put an arbitrary n − 1-dimensional space containing a tangent direction of this surface in its point B. Let us extend all directions in this space from point B geodesically and let us find the intersectional curve of the thus space Vn−1 obtained in this way with the surface V2 . We now have to decide whether all normal lines to that curve in point B are contained in Vn−1 . The answer to this question appears to be obvious, when we realize that the curve originated as the intersection of Vn−1 with V2 . Certainly every mathematician, not looking for the answer in a mathematical way but extending statements from Rn and Rn−1 to Vn and Vn−1 , would answer the question immediately in the affirmative. However, a rather complex calculation34 shows that the positive answer is correct only if Vn is of the special form Sn . Since Rn and Sn are of the same structure, it was to be expected that the conclusion from Rn to Sn provided an affirmative answer. I consider this example highly interesting because our pˇredstava clearly suggested a positive answer and yet the answer was incorrect. Finally, I will give an example for our remark about apparent exceptions to our proposition. Through each point B of the space Vn (n ≥ 2), we can pick n arbitrary directions and define (unit) lengths on them. The body thus emerging has a certain volume. We are asked whether the volume changes if point B moves along infinitesimally on a curve in Vn . For the case that Vn is a Euclidean space or a surface, the answer is clearly negative as has already been argued earlier through mathematical reasoning. If we extended our answer from Rn (or V2 ) to Sn , we would have to give a negative answer to that question, and this is confirmed by calculation. There are, however, such Vn that are not special spaces and for which this question must also be answered negatively. This, at first sight, seems to contradict our proposition. Under closer examination, we find that this case is precisely one where derived forms of the two tensors of curvature (of a plane, or Rn ) and Vn are equal. For a surface, we always have  λ

···λ Kωμλ = 0.

(C)

But this equation expresses precisely the condition that the volume of a given body is constant in Vn , if Kωμλν is its curvature tensor. It holds identically for Sn , confirming the answer given above. For general · · · λ of spaces, this condition is not always satisfied. But if it holds, then the derived expressions λ Kωμλ 34 Performed in the work of the author “Sur les courbes quasiasymptotiques” in the journal Christian Huygens, 1924–25.

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the curvature tensors are of the same form for a surface and the examined space (and in this case they all vanish), and therefore extending our conclusions from a surface to Vn is correct here as well and it is negative. The example teaches us about the—above-mentioned—caution in applying our proposition. We have to be cautious when extending our conclusions from the image of a surface to a space Vn which is not entirely general but special in some sense. These cases must always be decided by mathematical reasoning. ∗ Now several things to conclude. The motivation for this article was entirely mathematical, and it was inspired by the second example discussed above. But I think that it can also interest philosophers, since it is an attempt to find a mathematical basis for some functions of reasoning. I think that a mathematician well-versed in number theory would also be able to justify mathematically that we cannot imagine certain things, for example the irrational numbers. Also in modern geometry we find examples of an author arriving at results that he did not wish to find; a historical example is Beez (1875) who wanted to prove the absurdity of non-Euclidean geometry, but found a theorem about the non-deformability of a space Vn−1 in Rn (n > 3). References Beez, E.L.R., 1874. Ueber das Krümmungsmaass von Mannigfaltigkeiten höherer Ordnung. Math. Ann. 7, 387–395. Beez, E.L.R., 1875a. Ueber conforme Abbildung von Mannigfaltigkeiten höherer Ordnung. Z. Math. Phys. 20, 253–273. Beez, E.L.R., 1875b. Zur Theorie des Krümmungsmaasses von Mannigfaltigkeiten höherer Ordnung. Z. Math. Phys. 20, 423–444. Beez, E.L.R., 1876. Zur Theorie des Krümmungsmaasses von Mannigfaltigkeiten höherer Ordnung. Z. Math. Phys. 21, 373–401. Beez, E.L.R., 1879. Ueber das Riemannsche Krümmungsmass höherer Mannigfaltigkeiten. Z. Math. Phys. 24, 1–47. Beez, E.L.R., 1888. Über Euklidische und Nicht-Euklidische Geometrie. Wissenschaftliche Beilage aus dem Programm des Gymnasiums und Realgymnasiums zu Plauen i. V. Plauen. Berger, Marcel, 2003. A Panoramic View of Riemannian Geometry. Springer. Berwald, Ludwig, 1922. Zur Geometrie einer n-dimensionalen Riemannschen Mannigfaltigkeit im (n + 1)-dimensionalen Euklidisch-affinen Raum. Jahresber. Dtsch. Math.-Ver. 31, 162–170. Berwald, Ludwig, 1923. Differentialinvarianten in der Geometrie. Riemannsche Mannigfaltigkeiten und ihre Verallgemeinerungen. In: Enzyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, vol. 3, pp. 73–181. Blaschke, Wilhelm, Reidemeister, K., 1922. Über die Entwicklung der Affingeometrie. Jahresber. Dtsch. Math.-Ver. 31, 63–82. Cartan, Élie, 1923. Sur les variétés a connexion affine et la théorie de la relativité généralisée. Ann. Sci. Éc. Norm. Super. 40, 325–412. Cogliati, Alberto, 2014. Riemann’s Commentatio Mathematica, A Reassessment. Rev. Hist. Math. 20, 73–94. Cogliati, Alberto, 2016. Schouten, Levi-Civita and the notion of parallelism in Riemannian geometry. Hist. Math. 43, 427–443. Cogliati, Alberto, Mastrolia, Paolo, 2018. Cartan, Schouten and the search for connection. Hist. Math. 45, 39–74. Durnová, Helena, Kot˚ulek, Jan, Žádník, Vojtˇech, 2017. Václav Hlavatý (1894–1969). Cesta k jednotˇe. Masarykova univerzita, Brno. Durnová, Helena, Sauer, T., Geometry of Einstein’s unified field theory: the interactions between Václav Hlavatý and Albert Einstein. In preparation. Einstein, Albert, 2009. The Collected Papers of Albert Einstein, vol. 12. In: Kormos Buchwald, Diana, et al. (Eds.), The Berlin Years: Correspondence January–December 1921. Princeton University Press, Princeton.

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