Václav Hlavatý on intuition in Riemannian space. (English) Zbl 1428.01021
Authors’ 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.
Reviewer’s remarks: In the present paper, the main attention is given to the following items:
1. Scientific community in Prague in the early 1920s and Václav Hlavatý’s early career. In this section, a few biographical information about Hlavatý is provided and some attention is given to his personal situation when publishing his mathematical-philosophical paper “On intuition in Riemannian space” in the Czech philosophical journal Ruch filosofický, i.e., his cooperation with Czech and foreign scientists is described and some philosophical aspects of his research are noted.
2. The journal Ruch filosofický. Here, the development and state of philosophy of science in Czechoslovakia in 1918–1920 are considered. Also, reasons for founding the journal Ruch filosofický are noted.
3. Linguistic discussion. The relationships between some researches of the Czechoslovak scientists Hlavatý and Vorovka are described. A brief summary of Hlavatý’s article under consideration is given. In addition, some linguistic-philosophical comments on the title of the paper and the choice of terms for the article are discussed.
4. The argument of the paper. Section 5 is devoted to a more detailed consideration of Hlavatý’s article. Some overview to the mathematical argument laid out therein and an overview to the related researches of other scientists are given.
5. Concluding remarks. Here, the attention is given to analizing the fact that almost all of Hlavatý’s papers were published in mathematical journals but his paper “On intuition in Riemannian space” was published in the philosophical journal “Ruch filosofický”.
6. An English translation of Hlavatý’s paper by the authors is given in the appendix.
Reviewer’s remarks: In the present paper, the main attention is given to the following items:
1. Scientific community in Prague in the early 1920s and Václav Hlavatý’s early career. In this section, a few biographical information about Hlavatý is provided and some attention is given to his personal situation when publishing his mathematical-philosophical paper “On intuition in Riemannian space” in the Czech philosophical journal Ruch filosofický, i.e., his cooperation with Czech and foreign scientists is described and some philosophical aspects of his research are noted.
2. The journal Ruch filosofický. Here, the development and state of philosophy of science in Czechoslovakia in 1918–1920 are considered. Also, reasons for founding the journal Ruch filosofický are noted.
3. Linguistic discussion. The relationships between some researches of the Czechoslovak scientists Hlavatý and Vorovka are described. A brief summary of Hlavatý’s article under consideration is given. In addition, some linguistic-philosophical comments on the title of the paper and the choice of terms for the article are discussed.
4. The argument of the paper. Section 5 is devoted to a more detailed consideration of Hlavatý’s article. Some overview to the mathematical argument laid out therein and an overview to the related researches of other scientists are given.
5. Concluding remarks. Here, the attention is given to analizing the fact that almost all of Hlavatý’s papers were published in mathematical journals but his paper “On intuition in Riemannian space” was published in the philosophical journal “Ruch filosofický”.
6. An English translation of Hlavatý’s paper by the authors is given in the appendix.
Reviewer: Symon Serbenyuk (Kyiv)
MSC:
| 01A60 | History of mathematics in the 20th century |
| 00A30 | Philosophy of mathematics |
| 01A70 | Biographies, obituaries, personalia, bibliographies |
| 53-03 | History of differential geometry |
Keywords:
differential geometry; Riemannian geometry; intuition in mathematics; philosophy of mathematicsReferences:
| [1] | Beez, E. L.R., Ueber das Krümmungsmaass von Mannigfaltigkeiten höherer Ordnung, Math. Ann., 7, 387-395 (1874) · JFM 06.0466.03 |
| [2] | Beez, E. L.R., Ueber conforme Abbildung von Mannigfaltigkeiten höherer Ordnung, Z. Math. Phys., 20, 253-273 (1875) · JFM 07.0519.01 |
| [3] | Beez, E. L.R., Zur Theorie des Krümmungsmaasses von Mannigfaltigkeiten höherer Ordnung, Z. Math. Phys., 20, 423-444 (1875) · JFM 07.0306.02 |
| [4] | Beez, E. L.R., Zur Theorie des Krümmungsmaasses von Mannigfaltigkeiten höherer Ordnung, Z. Math. Phys., 21, 373-401 (1876) · JFM 09.0513.01 |
| [5] | Beez, E. L.R., Ueber das Riemannsche Krümmungsmass höherer Mannigfaltigkeiten, Z. Math. Phys., 24, 1-47 (1879) · JFM 09.0513.02 |
| [6] | Beez, E. L.R., Über Euklidische und Nicht-Euklidische Geometrie. Wissenschaftliche Beilage aus dem Programm des Gymnasiums und Realgymnasiums zu Plauen i. V (1888), Plauen · JFM 20.0512.01 |
| [7] | Berger, Marcel, A Panoramic View of Riemannian Geometry (2003), Springer · Zbl 1038.53002 |
| [8] | Berwald, Ludwig, Zur Geometrie einer \(n\)-dimensionalen Riemannschen Mannigfaltigkeit im \((n + 1)\)-dimensionalen Euklidisch-affinen Raum, Jahresber. Dtsch. Math.-Ver., 31, 162-170 (1922) · JFM 48.0846.04 |
| [9] | Berwald, Ludwig, Differentialinvarianten in der Geometrie. Riemannsche Mannigfaltigkeiten und ihre Verallgemeinerungen, (Enzyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, vol. 3 (1923)), 73-181 |
| [10] | Blaschke, Wilhelm; Reidemeister, K., Über die Entwicklung der Affingeometrie, Jahresber. Dtsch. Math.-Ver., 31, 63-82 (1922) · JFM 48.0804.01 |
| [11] | Cartan, Élie, Sur les variétés a connexion affine et la théorie de la relativité généralisée, Ann. Sci. Éc. Norm. Super., 40, 325-412 (1923) · JFM 49.0542.02 |
| [12] | Cogliati, Alberto, Riemann’s Commentatio Mathematica, A Reassessment, Rev. Hist. Math., 20, 73-94 (2014) · Zbl 1305.01027 |
| [13] | Cogliati, Alberto, Schouten, Levi-Civita and the notion of parallelism in Riemannian geometry, Hist. Math., 43, 427-443 (2016) · Zbl 1354.01013 |
| [14] | Cogliati, Alberto; Mastrolia, Paolo, Cartan, Schouten and the search for connection, Hist. Math., 45, 39-74 (2018) · Zbl 1397.01020 |
| [15] | Durnová, Helena; Kotůlek, Jan; Žádník, Vojtěch, Václav Hlavatý (1894-1969). Cesta k jednotě (2017), Masarykova univerzita: Masarykova univerzita Brno |
| [16] | Durnová, Helena, Sauer, T., Geometry of Einstein’s unified field theory: the interactions between Václav Hlavatý and Albert Einstein. In preparation.; Durnová, Helena, Sauer, T., Geometry of Einstein’s unified field theory: the interactions between Václav Hlavatý and Albert Einstein. In preparation. |
| [17] | Einstein, Albert, The Collected Papers of Albert Einstein, vol. 12, (Kormos Buchwald, Diana; etal., The Berlin Years: Correspondence January-December 1921 (2009), Princeton University Press: Princeton University Press Princeton) · Zbl 1173.01011 |
| [18] | Einstein, Albert; Grossmann, Marcel, Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation, Z. Math. Phys., 62, 225-261 (1913), Reprinted in The Collected Papers of Albert Einstein, vol. 4. The Swiss Years: Writings 1912-1914, M.J. Klein et al. (eds.), Princeton University Press, Princeton, 1995, pp. 294-343 · JFM 45.1114.02 |
| [19] | Giovanelli, Marco, ‘What is Truth?’ Einstein on Rods and Clocks in Relativity Theory (2017), University of Tübingen, Habilitation thesis |
| [20] | Goenner, Hubert, On the history of unified field theories, Living Rev. Relativ., 7 (2004), lrr-2004-2 · Zbl 1070.83024 |
| [21] | Goenner, Hubert, On the history of unified field theories. Part II. (ca. 1930-ca. 1965), Living Rev. Relativ., 17 (2014), lrr-2014-5 · Zbl 1320.83003 |
| [22] | Gray, Jeremy, Eduard Čech, Math. Intell., 16, 48-49 (1994) · Zbl 0811.01005 |
| [23] | Hlavatý, Václav, Les congruences dans les espaces non-euclidéens, Věstník KČSN, třída II, 30 (1923) · JFM 49.0541.02 |
| [24] | Hlavatý, Václav, Promítání z přímky na rovinu v prostoru čtyřrozměrném, ČPMF, 52, 250-272 (1923) · JFM 49.0488.01 |
| [25] | Hlavatý, Václav, Sur les courbes quasiasymptotiques de Bompiani, C. R. Acad. Sci., 178, 2041-2046 (1924) · JFM 50.0498.01 |
| [26] | Hlavatý, Václav, On názoru v prostoru Riemannově, Ruch filosofický, 4, 302-309 (1924) |
| [27] | Hlavatý, Václav, Promítání z přímky na rovinu v prostoru pětirozměrném, ČPMF, 53, 251-271 (1924) · JFM 50.0446.03 |
| [28] | Hlavatý, Václav, Sur les courbes quasiasymptotiques, Christiaan Huygens, 3, 209-245 (1923-1924) · JFM 50.0497.04 |
| [29] | Hlavatý, V., Diferenciální přímková geometrie, vol. I (1941), ČAVU: ČAVU Praha |
| [30] | Hlavatý, V., Diferenciální přímková geometrie, vol. II (1941), ČAVU: ČAVU Praha |
| [31] | Hlavatý, V., Projektivní geometrie I. Útvary jednoparametrické (1944), Melantrich: Melantrich Praha |
| [32] | Hlavatý, V., Projektivní geometrie II. Útvary dvojparametrické (1945), Melantrich: Melantrich Praha · Zbl 0061.30704 |
| [33] | Hlavatý, V., Differentielle Liniengeometrie (1945), Noordhoff: Noordhoff Groningen, (Translated by Max Pinl.) · Zbl 0063.02027 |
| [34] | Hlavatý, V., Differential Line Geometry (1953), Noordhoff: Noordhoff Groningen, (Translated from the German version by Harry Levy.) · Zbl 0051.39101 |
| [35] | Hlavatý, V., Geometry of Einstein’s Unified Field Theory (1958), Noordhoff: Noordhoff Groningen · Zbl 0082.21503 |
| [36] | Jastrzembská, Zdeňka, Spor o Einsteinovu teorii relativity v české filosofii první poloviny dvacátého století, Sborník prací filozofické fakulty brněnské univerzity, series B, 49, 137-149 (2002) |
| [37] | Katětov, Miroslav; Simon, Petr, The Mathematical Legacy of Eduard Čech (1993), Springer · Zbl 0871.54001 |
| [38] | Killing, Wilhelm, Die nicht-euklidischen Raumformen in analytischer Behandlung (1885), Teubner: Teubner Leipzig |
| [39] | Kobayashi, Shoshichi; Nomizu, Katsumi, Foundations of Differential Geometry, vol. 1 (1963), Interscience Publishers · Zbl 0091.34802 |
| [40] | Kobayashi, Shoshichi; Nomizu, Katsumi, Foundations of Differential Geometry, vol. 2 (1969), Interscience Publishers · Zbl 0091.34802 |
| [41] | Kowalewski, Gerhard, Bestand und Wandel (1950), R. Oldenbourg: R. Oldenbourg München · Zbl 0037.00203 |
| [42] | Kraus, Oskar, Fiktion und Hypothese in der Einsteinschen Relativitätstheorie, Ann. Philos., 2, 3, 335-396 (1921), 463-465 |
| [43] | Láska, Václav, O názoru a priori v matematice, Spisy přírodovědecké fakulty Karlovy university, 6 (1924) |
| [44] | Lehmkuhl, Dennis, Why Einstein did not believe that general relativity geometrizes gravity, Stud. Hist. Philos. Mod. Phys., 46, 316-326 (2014) · Zbl 1315.83007 |
| [45] | Lichnerowicz, A., Differential geometry and physical theories, (Perspectives in Geometry and Relativity — Essays in Honor of Václav Hlavatý (1966), IU Press: IU Press Bloomington, IN), 1-6 · Zbl 0168.19604 |
| [46] | Nožička, František, Profesor Václav Hlavatý, český matematik světového jména, Časopis pro pěstování matematiky, 94, 374-380 (1969) |
| [47] | Pavlincová, Helena, Karel Vorovka. Cesta matematika k filosofii (2010), Filosofia: Filosofia Prague |
| [48] | Pelikán, František, Vláda demokracie ve filosofii, Ruch filosofický, 1, 1-5 (1921) |
| [49] | Reich, Karin, Levi-Civitasche Parallelverschiebung, affiner Zusammenhang, Uebertragungsprinzip: 1916/17-1922/23, Arch. Hist. Exact Sci., 44, 77-105 (1992) · Zbl 0767.01023 |
| [50] | Reichenbach, Hans, Die Philosophie der Raum-Zeit-Lehre (1928), de Gruyter: de Gruyter Berlin, Leipzig · JFM 54.0937.17 |
| [51] | Ricci, Gregorio; Levi-Civita, Tullio, Méthodes de calcul différentiel absolu et leurs applications, Math. Ann., 54, 125-201 (1901) · JFM 31.0297.01 |
| [52] | Rowe, David E., Dirk Jan Struik and His contributions to the History of Mathematics, Hist. Math., 21, 245-273 (1994) · Zbl 0827.01016 |
| [53] | Schoute, P. H., Mehrdimensionale Geometrie I. Band. Die linearen Räume (1902), Göschen: Göschen Leipzig · JFM 33.0571.03 |
| [54] | Schouten, Jan Arnoldus, Über die verschiedenen Arten der Übertragung in einer \(n\)-dimensionalen Mannigfaltigkeit, die einer Differentialgeometrie zugrunde gelegt werden können, Math. Z., 13, 56-81 (1922) · JFM 48.0858.01 |
| [55] | Schouten, Jan Arnoldus, Nachtrag zur Arbeit über die verschiedenen Arten der Übertragung, Math. Z.. Math. Z., Math. Z., 15, 168-81 (1922) · JFM 48.0858.02 |
| [56] | Schouten, Jan Arnoldus, Über die Einordnung der Affingeometrie in die Theorie der höheren Übertragungen, Math. Z., 17, 161-182 (1923) · JFM 49.0541.03 |
| [57] | Schouten, Jan Arnoldus, Über die Einordnung der Affingeometrie in die Theorie der höheren Übertragungen. II, Math. Z., 17, 183-188 (1923) · JFM 49.0541.03 |
| [58] | Schouten, Jan Arnoldus, Der Ricci-Kalkül. Eine Einführung in die neueren Methoden und Probleme der mehrdimensionalen Differentialgeometrie (1924), Springer: Springer Berlin · JFM 50.0588.01 |
| [59] | Schouten, Jan Arnoldus, Neue Gesichtspunkte zur Grundlegung der Differentialgeometrie, Jahresber. Dtsch. Math.-Ver., 33, 89-91 (1925) · JFM 51.0442.04 |
| [60] | Schur, F., Ueber den Zusammenhang der Räume constanten Riemann’schen Krümmungsmaasses mit den projectiven Räumen, Math. Ann., 27, 4, 537-567 (1886) · JFM 18.0713.02 |
| [61] | Struik, Dirk J., Grundzüge der mehrdimensionalen Differentialgeometrie in direkter Darstellung (1922), Julius Springer: Julius Springer Berlin · JFM 48.0841.04 |
| [62] | Struik, Dirk J., Schoute, Pieter Hendrik, Complete Dictionary of Scientific Biography. Encyclopedia.com. 8 Dec (2017) |
| [63] | Truesdell, C., Václav Hlavatý, Int. Math. News, 29/30, 12, 2-4 (1953) |
| [64] | Volkert, Klaus, Das Undenkbare denken. Die Rezeption der nichteuklidischen Geometrie im deutschsprachigen Raum (1860-1900) (2013), Springer · Zbl 1288.01003 |
| [65] | Volkert, Klaus, In höheren Räumen. Der Weg der Geometrie in die vierte Dimension (2018), Springer · Zbl 1395.01008 |
| [66] | Vorovka, Karel, O názoru v matematice (1917), ČAVU: ČAVU Prague |
| [67] | Vorovka, Karel, Věda a filosofie, Ruch filosofický, 1, 5-10 (1921) |
| [68] | de Vries, H., Die Lehre von der Zentralprojektion im vierdimensionalen Raume (1905), Göschen: Göschen Leipzig · JFM 35.0543.05 |
| [69] | Weyl, Hermann, Reine Infinitesimalgeometrie, Math. Z., 2, 384-411 (1918) · JFM 46.1301.01 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
